We thank Marco Manetti for stimulating discussions on the subject
and for useful comments and suggestions on the first version of this paper; d.i. thanks the Mathematical Department
“Guido Castelnuovo”, Sapienza Università di Roma for
the hospitality.
6pt
Infinitesimal deformations and sheaves of DGLAs
In this section, we study infinitesimal deformations of a coherent
sheaf \(\mathcal {F}\)  of \(\mathcal {O}_X\) -modules on a smooth projective
variety \(X\)  and explain
how these deformations can be naturally described in terms of a
sheaf of differential graded Lie algebras on \(X\) .
An infinitesimal deformation of the coherent sheaf
of \(\mathcal {O}_X\) -modules \(\mathcal {F}\)  over \(A\in \bf {Art}_{\mathbb {K}}\)  is given by a
coherent sheaf \(\mathcal {F}_A\)  of \(\mathcal {O}_X\otimes A\) -modules on
\(X\times \operatorname{Spec}A\) , flat over \(A\) , with a morphism of sheaves \(\pi :\mathcal {F}_A \rightarrow \mathcal {F}\)  inducing an isomorphism
\(\mathcal {F}_A\otimes _A \mathbb {K}\cong \mathcal {F}\) .
Two deformations \(\mathcal {F}_A, \mathcal {F^{\prime }}_A\)  of the coherent
sheaf \(\mathcal {F}\)  over \(A\)  are isomorphic if there exists an
isomorphism of sheaves \(f: \mathcal {F}_A \rightarrow \mathcal {F^{\prime }}_A\) , that
commutes with the morphisms to \(\mathcal {F}\) .
We denote by \(\operatorname{Def}_{\mathcal {F}}: \bf {Art}_{\mathbb {K}}\rightarrow \bf {Set}\)  the functor
of infinitesimal deformations of the sheaf \(\mathcal {F}\) .
We start by studying infinitesimal deformations of a coherent
sheaf \({\mathcal {F}}\)  of \(\mathcal {O}_X\) -modules on an affine variety \(X\) .
Let \(X=\operatorname{Spec}R\) , where \(R\)  is a Noetherian \(\mathbb {K}\) -algebra and let
\(\mathcal {F}\)  be the coherent sheaf associated with a finitely generated
\(R\) -module \(M\) ; in this simple case, deformations of the sheaf
\(\mathcal {F}\)  reduce to deformations of the \(R\) -module \(M\) .
An infinitesimal deformation of the \(R\) -module \(M\)  over \(A\in \bf {Art}_{\mathbb {K}}\)  is given by a \(R\otimes A\) -module \(M_A\) , flat over \(A\) , with a morphism \(\pi : M_A \rightarrow M\)  inducing an isomorphism
\(M_A\otimes _A \mathbb {K}\cong M\) .
Two deformations \(M_A\)  and \( M^{\prime }_A\)  of the module \(M\)  over \(A\)  are isomorphic if there exists an
isomorphism of \(R\otimes A\) -modules \(f: M_A \rightarrow M^{\prime }_A\) , that
commutes with the morphisms to \(M\) .
Next, let
\( \cdots \stackrel{d}{\longrightarrow } R^{n_{1}}\stackrel{d}{\longrightarrow } R^{n_0}\stackrel{d}{\longrightarrow } M \longrightarrow 0\) 
be a presentation of \(M\)  as \(R\) -module.
If \(M_A\)  is a deformation of \(M\)  over \(A\) , then it is an \(A\) -flat \(R\otimes A\) -module; therefore, flatness allows to lift relations between generators and to construct the exact sequence
\(\cdots \stackrel{d_A}{\longrightarrow } R^{n_{1}}\otimes A\stackrel{d_A}{\longrightarrow } R^{n_0}\otimes A\stackrel{d_A}{\longrightarrow } M_A \longrightarrow 0, \) 
that reduces to (REF ) when tensored by \(\mathbb {K}\)  over \(A\) .
On the other hand, the datum of such an exact sequence assures flatness of the \(R\otimes A\) -module \(M_A\)  and so it defines a deformation of \(M\)  over \(A\)  (see [1]}, or [2]} for details of these correspondences).
Moreover, if \(M_A\)  and \(M^{\prime }_A\)  are isomorphic deformations of \(M\)  over \(A\) , the isomorphism between them lifts to an isomorphism between the correspondent deformed complexes and viceversa.
Next, we return to the global case of a coherent sheaf \(\mathcal {F}\)  of \(\mathcal {O}_X\) -modules on a smooth projective variety \(X\) .
Let
\(0 \longrightarrow \mathcal {E}^{-m} \stackrel{d}{\longrightarrow }\cdots \stackrel{d}{\longrightarrow } \mathcal {E}^{-1}\stackrel{d}{\longrightarrow } \mathcal {E}^0\stackrel{d}{\longrightarrow } \mathcal {F} \longrightarrow 0 \) 
be a global syzygy for \({\mathcal {F}}\) , and denote by \(\mathcal {E}^{\cdot }\)  the complex of
locally free sheaves
\( (\mathcal {E}^{\cdot },d): \qquad \qquad 0\longrightarrow \mathcal {E}^{-m}\stackrel{d}{\longrightarrow } \cdots \stackrel{d}{\longrightarrow }\mathcal {E}^{-1} \stackrel{d}{\longrightarrow }\mathcal {E}^0 \longrightarrow 0.\) 
Let \(\mathcal {U}=\lbrace  U_i \rbrace _{i\in I}\)  be an affineor Stein, if we work in the complex analytic category. open cover of \(X\) , such
that every sheaf of \(\mathcal {E}^{\cdot }\)  is free on each \(U_i\) .
The Kodaira-Spencer approach to infinitesimal deformations of
\(\mathcal {F}\)  consists in deforming the sheaf \(\mathcal {F}\)  locally
in such a way that local deformations glue together to a global
sheaf, or equivalently, in view of the above discussion of the
affine case, in deforming the complex \((\mathcal {E}^{\cdot },d)\)  on every
open set \(U_i\)  in such a way that these data glue together in
cohomology.
Following this approach, let us make explicit the deformation data:
the first datum is an element \(l=\lbrace l_i\rbrace _i\in \prod _i {\mathcal {E}nd}^1(\mathcal {E}^{\cdot })(U_i)\otimes \mathfrak {m}_A\)  defining, on every open set \(U_i\) , a complex
\((\mathcal {E}^{\cdot }|_{U_i}\otimes A,d + l_i)\)  which is a deformation of the complex
\((\mathcal {E}^{\cdot }|_{U_i},d)\) . Note that the condition for \((\mathcal {E}^{\cdot }|_{U_i}\otimes A,d + l_i)\)  to be a complex
is the Maurer-Cartan equation:
\(dl_i+\frac{1}{2}[l_i,l_i]=0, \quad \mbox{for all } i\in I.\) 
Also note that, by upper semicontinuity of cohomology, the complex \((\mathcal {E}^{\cdot }|_{U_i}\otimes A,d + l_i)\)  is exact except possibly at
zero level.
To glue together the deformed local complexes \((\mathcal {E}^{\cdot }|_{U_i}\otimes A,d + l_i)\) , we need to specify isomorphisms
between the deformed complexes on the double intersections of open
sets of the cover \({\mathcal {U}}\) . Since these isomorphisms will have to be deformations of the identity, they will be of
the form
\(e^{m_{ij}}: (\mathcal {E}^{\cdot }|_{U_{ij}}\otimes A,d + l_j) \rightarrow (\mathcal {E}^{\cdot }|_{U_{ij}}\otimes A,d + l_i),\) 
with
\(m=\lbrace m_{ij}\rbrace _{i<j}\in \prod _{i<j} {\mathcal {E}nd}^0(\mathcal {E}^{\cdot })(U_{ij})\otimes \mathfrak {m}_A\) . The compatibiliy with the differentials, i.e., the commutativity of the diagrams
\( {\mathcal {E}^{\cdot }|_{U_{ij}}\otimes A [r]^{e^{m_{ij}}}[d]_{d+l_j|_{U_{ij}}} &\mathcal {E}^{\cdot }|_{U_{ij}}\otimes A[d]^{d+l_i|_{U_{ij}}}\\\mathcal {E}^{\cdot }|_{U_{ij}}\otimes A [r]^{e^{m_{ij}}} &\mathcal {E}^{\cdot }|_{U_{ij}}\otimes A}\) 
can be written as
\(d+l_i|_{U_{ij}}=e^{m_{ij}}(d+l_j|_{U_{ij}})e^{-m_{ij}}\) , i.e., as
\( l_i|_{U_{ij}}=e^{m_{ij}}*l_j|_{U_{ij}}, \quad \mbox{for all } i<j. \) 
Finally, the above isomorphisms have to satisfy the cocycle condition up to homotopy.
Indeed, in order to obtain a deformation of \({\mathcal {F}}\) ,
we actually do not want to glue together the complexes \((\mathcal {E}^{\cdot }|_{U_i}\otimes A,d + l_i)\) , but rather their cohomology sheaves. In other words, we require
\(e^{m_{jk}} e^{-m_{ik}} e^{m_{ij}}\)  to be
homotopic to the identity on triple intersections. Taking logarithm, what we require is that
\(m_{jk} \bullet -m_{ik} \bullet m_{ij}\)  is homotopy equivalent to
zero, i.e.,
\(m_{jk} |_{U_{ijk}} \bullet - m_{ik}|_{U_{ijk}} \bullet m_{ij}|_{U_{ijk}} =[d + l_j |_{U_{ijk}} , n_{ijk}],\) 
for some \(n=\lbrace n_{ijk}\rbrace _{ijk} \in \prod _{i<j<k} {\mathcal {E}nd}^{-1}(\mathcal {E}^{\cdot })(U_{ijk})\) .
This homotopy cocycle equation is conveniently rewritten as
\( m_{jk} |_{U_{ijk}} \bullet - m_{ik}|_{U_{ijk}} \bullet m_{ij}|_{U_{ijk}} =d_{\mathcal {E}nd^{*}(\mathcal {E}^{\cdot })}n_{ijk}+[l_j |_{U_{ijk}} , n_{ijk}].\) 
Next, let explain how the data introduced above are concretely
linked with deformations of the coherent sheaf \(\mathcal {F}\)  over
\(A\) . As the homotopy cocycle equation is satisfied, the local
\(A\) -flat sheaves of \(\mathcal {O}_X|_{U_i}\otimes A\) -modules \({\mathcal {F}}_{A,U_i}:={\mathcal {H}}^*(\mathcal {E}^{\cdot }|_{U_i}\otimes A,d + l_i)\) 
glue together to give a global coherent sheaf \({\mathcal {F}}_A\) 
which is a deformation of \({\mathcal {F}}\) . On the other hand, every
deformation \({\mathcal {F}}_A\)  of the sheaf \(\mathcal {F}\)  can be
obtained in this way. Indeed, the resolution \((\mathcal {E}^{\cdot }, d)\) 
locally extends to projective resolutions
\((\mathcal {E}^{\cdot }\vert _{U_i}\otimes A, d+l_i)\)  of \({\mathcal {F}}_A\vert _{U_i}\) ; these deformed local resolutions are linked each
other on double intersections by isomorphisms of complexes lifting
the identity of \({\mathcal {F}}_A\)  and the compositions of these
isomorphisms on triple intersections are homotopy to the identity,
since they lift the identity of \({\mathcal {F}}_A\)  and liftings are
unique up to homotopy.
Let now \({\mathcal {F}}_A\)  and \({\mathcal {F}^{\prime }}_A\)  be isomorphic
deformations of the sheaf \({\mathcal {F}}\) , associated with deformation data \((l,m)\) 
and \((l^{\prime },m^{\prime })\) , respectively. The restriction to every open set
\(U_i\)  of the isomorphism between \({\mathcal {F}}_A\)  and \({\mathcal {F}^{\prime }}_A\)  lifts to local isomorphisms
between the correspondent deformed complexes. Since these isomorphisms
specialize to identities of \((\mathcal {E}^\cdot \vert _{U_i},d)\) , they are of the form \(e^{a_{i}}:(\mathcal {E}^\cdot |_{U_i}\otimes A, d+l_i) \rightarrow (\mathcal {E}^\cdot |_{U_i}\otimes A, d+l^{\prime }_i)\) ,
where \(a=\lbrace a_i\rbrace _i \in \prod _i {\mathcal {E}nd}^0(\mathcal {E}^{\cdot })(U_i)\otimes \mathfrak {m}_A\) . As above, compatibility with the differentials translates into the equations
\(e^{a_i} * l_i=l^{\prime }_i, \quad \mbox{for all } i\in I. \) 
Finally, since the local isomorphisms \(e^{a_i}\)  lift a global isomorphism in cohomology, the diagrams
\( {(\mathcal {E}^{\cdot }|_{U_{ij}}\otimes A, d+l_j\vert _{U_{ij}})[r]^{e^{m_{ij}}}[d]_{e^{a_j}|_{U_{ij}}} &(\mathcal {E}^{\cdot }|_{U_{ij}}\otimes A, d+l_i\vert _{U_{ij}})[d]^{e^{a_i}|_{U_{ij}}}\\(\mathcal {E}^{\cdot }|_{U_{ij}}\otimes A, d+l^{\prime }_j\vert _{U_{ij}}) [r]^{e^{m^{\prime }_{ij}}} &(\mathcal {E}^{\cdot }|_{U_{ij}}\otimes A, d+l^{\prime }_i\vert _{U_{ij}}),}\) 
expressing compatibility with the gluing morphisms, commute in cohomology. Moreover, since the compositions
\(e^{-m_{ij}} e^{-a_i} e^{ m^{\prime }_{ij}} e^{a_j}\)  lift the identity of \({\mathcal {F}}_A\)  on double intersections and liftings are unique up to homotopy, these compositions are homotopy to identity and, reasoning as above, we find
\( -m_{ij} \bullet -a_i|_{U_{ij}} \bullet m^{\prime }_{ij} \bullet a_j|_{U_{ij}} = d_{\mathcal {E}nd^{*}(\mathcal {E}^{\cdot })}b_{ij}+ [l_j|_{U_{ij}}, b_{ij}],\) 
for some \(b=\lbrace b_{ij}\rbrace _{i<j} \in \prod _{i<j} {\mathcal {E}nd}^{-1}(\mathcal {E}^{\cdot })(U_{ij})\otimes \mathfrak {m}_A\) .
Viceversa, if for the deformation data \((l,m)\)  and \((l^{\prime },m^{\prime })\)  there exist \(a=\lbrace a_i\rbrace _i \in \prod _i {\mathcal {E}nd}^0(\mathcal {E}^{\cdot })(U_i)\otimes \mathfrak {m}_A\)  and \(b=\lbrace b_{ij}\rbrace _{i<j} \in \prod _{i<j} {\mathcal {E}nd}^{-1}(\mathcal {E}^{\cdot })(U_{ij})\otimes \mathfrak {m}_A\)  that satisfy equations above, the local isomorphisms \(e^{a_i}\)  glue together in cohomology to give a global isomorphism of the correspondent deformed sheaves \({\mathcal {F}}_A\)  and \({\mathcal {F}^{\prime }}_A\) .
Summing up, we have shown that in the Kodaira-Spencer approach,
infinitesimal deformations of the coherent sheaf \({\mathcal {F}}\) 
are controlled by the sheaf of DGLAs \({\mathcal {E}nd}^*(\mathcal {E}^{\cdot })\) ,
via the equations above. At the end of Section , we
will apply techniques of semicosimplicial DGLAs developed in this
paper to recover the classical well known fact that the functor of
infinitesimal deformations of \(\mathcal {F}\)  has \(\operatorname{Ext}^1({\mathcal {F}},{\mathcal {F}})\)  as tangent space and its obstructions are
contained in \(\operatorname{Ext}^2({\mathcal {F}},{\mathcal {F}})\) .
Remark 1.1 The above description of the functor of infinitesimal deformations of \({\mathcal {F}}\)  is actually independent of the resolution chosen. Indeed, the DGLAs of the
endomorphisms of any two locally free resolutions of \({\mathcal {F}}\)  are
quasi-isomorphic (see,e.g., [3]}).

Remark 1.2 If the sheaf \(\mathcal {F}\)  is locally free, then we can take its
trivial resolution \(0\rightarrow \mathcal {F}\rightarrow \mathcal {F}\rightarrow 0\) ; thus, we
recover the well known fact that the infinitesimal deformations
of \(\mathcal {F}\)  are controlled by the sheaf \(\mathcal {E}nd (\mathcal {F})\)  of the endomorphism of
\(\mathcal {F}\)  , via the Čech functor \(H^1(X,\mathcal {E}nd (\mathcal {F}))\) .

Remark 1.3 Note that the results of this section actually hold under the hypotesis that \({\mathcal {F}}\)  admists a global syzygy. This hypothesis is always satisfied, but in the general case the resolution is less obvious. Indeed, following Illusie [4]}, for any sheaf \(\mathcal {F}\)  of \({\mathcal {O}}_X\) -modules on a topological space \(X\) , one can construct the standard free resolution of \(\mathcal {F}\) :
\( \ldots \longrightarrow {\mathcal {R}}({\mathcal {F}})^{2} \stackrel{D^2}{\longrightarrow } {\mathcal {R}}({\mathcal {F}})^{1} \stackrel{D^1}{\longrightarrow } {\mathcal {R}}({\mathcal {F}})^{0} \longrightarrow \mathcal {F} \longrightarrow 0.\) 
Its terms are defined by recurrence: \({\mathcal {R}}({\mathcal {F}})^{0}\)  is the free sheaf of \(\mathcal {O}_X\) -modules associated with the presheaf \(U\mapsto \mathcal {O}_X(U)^{{\mathcal {F}}(U)}\) , given on every open set \(U\subset X\)  by the free \(\mathcal {O}_X(U)\) -module generated by \({\mathcal {F}}(U)\) ; \({\mathcal {R}}({\mathcal {F}})^{j}\)  is the free sheaf of \(\mathcal {O}_X\) -modules associated with the presheaf \(U\mapsto \mathcal {O}_X(U)^{{\mathcal {R}}({\mathcal {F}})^{j-1}(U)}\) , given on every open set \(U\subset X\)  by the free \(\mathcal {O}_X(U)\) -module generated by \({\mathcal {R}}({\mathcal {F}})^{j-1}(U)\) .
To define morphisms \(D^j\) , let's write explicitly elements in \({\mathcal {R}}({\mathcal {F}})^{j}(U)\) .
An element in \({\mathcal {R}}({\mathcal {F}})^{0}(U)\)  is of the form \(a^{i_0} \odot f_{i_0}\) , where \(a^{i_0}\in \mathcal {O}_X(U)\) , \(f_{i_0}\in \mathcal {F} (U)\) , and we used the \(\odot \)  to denote the action of \({\mathcal {O}}_X(U)\)  on the free \(\mathcal {O}_X(U)\) -module generated by \( \mathcal {F} (U)\) , in order to distinguish it from the action of \({\mathcal {O}}_X(U)\)  on the \(\mathcal {O}_X(U)\) -module \( \mathcal {F} (U)\) .
Recursively,
an element in \({\mathcal {R}}({\mathcal {F}})^{j}(U)\)  is of the form
\(a^{i_j}\odot a_{i_j}^{i_{j-1}} \odot \cdots \odot a^{i_0}_{i_1}\odot f_{i_0}\) 
where \(a_{i_k}^{i_{k-1}}\in {\mathcal {O}}_X(U)\) , \(f_{i_0}\in \mathcal {F} (U)\) .
The differential of the resolution is defined as \(D^j=\sum _{k=0}^{j}(-1)^i d^j_k\) , where \(d^j_k:{\mathcal {R}}({\mathcal {F}})^{j} \longrightarrow {\mathcal {R}}({\mathcal {F}})^{j-1}\)  is defined by
\(a^{i_j}\odot \cdots \odot a_{i_{k+1}}^{i_{k}}\odot a_{i_k}^{i_{k-1}}\odot \cdots \odot a^{i_0}_{i_1}\odot f_{i_0}\mapsto a^{i_j} \odot \cdots \odot a_{i_{k+1}}^{i_{k}} a_{i_k}^{i_{k-1}}\odot \cdots \odot a^{i_0}_{i_1}\odot f_{i_0}\) 
The relevant fact is that the sequence of free sheaves of \(\mathcal {O}_X\) -modules \(({\mathcal {R}}({\mathcal {F}})^{\cdot }, D^\cdot ) \rightarrow \mathcal {F}\)  is a resolution of \(\mathcal {F}\)  [4]}. This construction can be done for every sheaf \(\mathcal {F}\)  of \(\mathcal {O}_X\) -modules on a topological space \(X\) ; Illusie obtains it as an example of the even more general construction of the standard simplicial resolution of a pair of adjont functors [4]}.


Semicosimplicial DGLAs and the functor \(H^1_{\rm sc}(\exp {\mathfrak {g}}^\Delta )\)
A semicosimplicial differential graded Lie algebra is a
covariant functor \(\mathbf {\Delta }_{\operatorname{mon}}\rightarrow \mathbf {DGLA}\) , from the category
\(\mathbf {\Delta }_{\operatorname{mon}}\) , whose objects are finite
ordinal sets and whose morphisms are order-preserving injective
maps between them, to the category of DGLAs. Equivalently, a
semicosimplicial DGLA \({\mathfrak {g}}^\Delta \)  is a diagram
\({ {{\mathfrak {g}}_0}@<2pt>[r]@<-2pt>[r] & { {\mathfrak {g}}_1}@<4pt>[r] [r] @<-4pt>[r] & { {\mathfrak {g}}_2}@<6pt>[r] @<2pt>[r] @<-2pt>[r] @<-6pt>[r]&\cdots },\) 
where each \({\mathfrak {g}}_i\)  is a DGLA, and for each
\(i>0\)  there are \(i+1\)  morphisms of DGLAs
\(\partial _{k,i}\colon {\mathfrak {g}}_{i-1}\rightarrow {\mathfrak {g}}_{i},\qquad k=0,\dots ,i,\) 
such that
\(\partial _{k+1,i+1}\partial _{l,i}=\partial _{l,i+1}\partial _{k,i}\) ,
for any
\(k\ge l\) .
A classical example is the following: given a sheaf \({\mathcal {L}}\)  of DGLAs on a topological space \(X\) , and an open cover \({\mathcal {U}}\)  of \(X\) , one has the Čech cosimplicial DGLA \({\mathcal {L}}({\mathcal {U}})\) ,
\({ {\prod _i\mathcal {L}(U_i)}@<2pt>[r]@<-2pt>[r] & {\prod _{i<j}\mathcal {L}(U_{ij})}@<4pt>[r] [r] @<-4pt>[r] &{\prod _{i<j<k}\mathcal {L}(U_{ijk})}@<6pt>[r] @<2pt>[r] @<-2pt>[r] @<-6pt>[r]& \cdots },\) 
where the morphisms \(\partial _{k,i}\)  are the restriction maps.
Definition 2.1 Let \(\mathfrak {g}^{\Delta }\)  be a semicosimplicial DGLA. The functor
\(Z^1_{\rm sc}(\exp \mathfrak {g}^{\Delta }) :\mathbf {Art}_{\mathbb {K}} \rightarrow \mathbf {Set}\) 
is defined, for all
\(A\in \mathbf {Art}_{\mathbb {K}}\) , by
\( Z^1_{\rm sc}(\exp \mathfrak {g}^{\Delta })(A)= \left\lbrace  (l,m)\in (\mathfrak {g}_0^1 \oplus \mathfrak {g}^0_1) \otimes \mathfrak {m}_A \left|\begin{array}{l}dl+\frac{1}{2}[l,l]=0,\\ \partial _{1,1}l=e^{m}*\partial _{0,1}l, \\{\partial _{0,2}m} \bullet {-\partial _{1,2}m} \bullet {\partial _{2,2}m} =dn+[\partial _{2,2}\partial _{0,1}l,n]\\\qquad \qquad \qquad \qquad \text{for some $n\in {\mathfrak {g}}_2^{-1}\otimes {\mathfrak {m}}_A$}\end{array} \right.\right\rbrace .\) 

Remark 2.2 
In DGLA theory, given a DGLA \(L\)  and a Maurer-Cartan element \(x\)  in \(\operatorname{MC}_L (A)\) , the set
\( {\rm Stab}(x)= \lbrace dh+[x,h] \mid h \in L^{-1}\otimes {\mathfrak {m}}_A\rbrace \) 
is called the irrelevant stabilizer of \(x\) . Note that
\({\rm Stab}(x) \subseteq {\rm stab}(x)\) , where
\({\rm stab}(x)= \lbrace  a \in L^0\otimes \mathfrak {m}_A \mid e^a*x=x \rbrace \) 
is the stabilizer of \(x\)  under the gauge action of \(L^0\otimes \mathfrak {m}_A\)  on \(\operatorname{MC}_L (A)\) .
Also note that,
for any \(a \in L^0\otimes \mathfrak {m}_A\) ,
\(e^a e^{{\rm Stab}(x)} e^{-a}= e^{{\rm Stab}(y)}\) , with
\(y=e^a*x.\)

We now introduce an equivalence relation on the set \( Z^1_{\rm sc}(\exp \mathfrak {g}^{\Delta })(A)\)  as follows: we say that two elements
\((l_0,m_0)\)  and \((l_1,m_1) \in Z^1_{\rm sc} (\exp \mathfrak {g}^{\Delta })(A)\) 
are equivalent under the relation \(\sim \)  if and only if there
exist elements \(a \in \mathfrak {g}^0_0\otimes \mathfrak {m}_A\)  and \(b\in {\mathfrak {g}}_1^{-1}\otimes {\mathfrak {m}}_A\)  such that
\({\left\lbrace \begin{array}{ll}e^a * l_0=l_1\\- m_0\bullet -\partial _{1,1}a \bullet m_1\bullet \partial _{0,1}a=db+[\partial _{0,1}l_0,b].\end{array}\right.}\) 
Remark 2.3 The relation \(\sim \)  is actually an equivalence relation on
\(Z^1_{\rm sc}(\exp \mathfrak {g}^{\Delta })(A)\) . First note that the set
\(Z^1_{\rm sc}(\exp \mathfrak {g}^{\Delta })(A)\)  is closed under \(\sim \) .
Indeed, let \((l_0,m_0)\)  and \((l_1,m_1) \in (\mathfrak {g}_0^1\oplus \mathfrak {g}_1^0)\otimes \mathfrak {m}_A\)  be equivalent under \(\sim \)  via elements \(a \in \mathfrak {g}^0_0\otimes \mathfrak {m}_A\)  and \(b\in {\mathfrak {g}}_1^{-1}\otimes {\mathfrak {m}}_A\) , and suppose that \((l_0,m_0) \in Z^1_{\rm sc} (\exp \mathfrak {g}^{\Delta })(A)\) . Then \(l_1=e^a*l_0\)  satisfies the Maurer Cartan
equation and
\( e^{m_1}*\partial _{0,1}l_1= e^{ \partial _{1,1}a \bullet m_0 \bullet (db+[\partial _{0,1}l_0,b])\bullet -\partial _{0,1}a} e^{\partial _{0,1}a }* \partial _{0,1} l_0= e^{\partial _{1,1}a }* \partial _{1,1} l_0 = \partial _{1,1}l_1. \) 
Moreover, an easy calculation, using relations between maps \(\partial _{j,k}\)  and Remark REF , shows that
\({\partial _{0,2}m_1} \bullet {-\partial _{1,2}m_1} \bullet {\partial _{2,2}m_1}\) 
is an element of the irrelevant stabilizer of \(\partial _{2,2}\partial _{0,1}l_1\) .
Secondly \(\sim \)  is an equivalent relation.
Reflexivity is trivial; for simmetry,
let \((l_0,m_0)\)  and \((l_1,m_1)\)  be equivalent via elements
\(\ a \in \mathfrak {g}^0_0\otimes \mathfrak {m}_A\)  and \(b \in {\mathfrak {g}}_1^{-1}\otimes {\mathfrak {m}}_A\) , then \(e^{-a} * l_1=l_0\)  and
\( - m_1\bullet \partial _{1,1}(a) \bullet m_0 \bullet -\partial _{0,1}(a)= \partial _{0,1}(a) \bullet -(db+[\partial _{0,1}l_0,b]) \bullet -\partial _{0,1}(a)\)  is an element of the irrelevant stabilizer of \(\partial _{0,1}l_1\) , by Remark REF .
Next, let \((l_0,m_0)\sim (l_1,m_1)\)  via \(a \in \mathfrak {g}^0_0\otimes \mathfrak {m}_A\)  and \(b\in {\mathfrak {g}}_1^{-1}\otimes {\mathfrak {m}}_A \) , and \((l_1,m_1) \sim (l_2,m_2)\)  via \(\alpha \in \mathfrak {g}^0_0\otimes \mathfrak {m}_A\)  and \(\beta \in {\mathfrak {g}}_1^{-1}\otimes {\mathfrak {m}}_A \) ;
then, \(e^{\alpha \bullet a}*l_0=l_2\)  and
\(- m_0\bullet \partial _{1,1}(-(b\bullet a)) \bullet m_2 \bullet \partial _{0,1}(b\bullet a) = - m_0\bullet - \partial _{1,1} (a) \bullet - \partial _{1,1} (b) \bullet m_2 \bullet \partial _{0,1} (b) \bullet \partial _{0,1} (a) =\) 
\(- m_0\bullet - \partial _{1,1}(a) \bullet m_1\bullet (db+[\partial _{0,1}l_0,b]) \bullet \partial _{0,1}(a),\) 
by Remark REF , it is an element of the irrelevant stabilizer of \(\partial _{0,1}l_0\) , therefore \(\sim \)  is transitive.

Definition 2.4 Let \(\mathfrak {g}^\Delta \)  be a semicosimplicial DGLA, the functor
\( H^1_{\rm sc}(\exp \mathfrak {g}^\Delta ) : \mathbf {Art}_{\mathbb {K}}\rightarrow \mathbf {Set}\) 
is defined, for all \(A\in \mathbf {Art}_{\mathbb {K}}\) , by
\(H^1_{\rm sc}(\exp \mathfrak {g}^\Delta )(A)= \frac{Z^1_{\rm sc}(\exp \mathfrak {g}^\Delta )(A)}{\sim }.\) 

Remark 2.5 Note that, if \(\mathfrak {g}^\Delta \)  is a semicosimplicial Lie
algebra, i.e., if all the DGLAs \({\mathfrak {g}}_i\)  are concentrated in degree zero, then the functor
\(H^1_{\rm sc}(\exp \mathfrak {g}^\Delta )\)  reduces to
the one defined in [7]}.

Lemma 2.6 The projection \( \pi : Z^1_{\rm sc} (\exp \mathfrak {g}^{\Delta })\longrightarrow H^1_{\rm sc}(\exp \mathfrak {g}^\Delta )\)  is a
smooth morphism of functors.

Let \( \beta : B \longrightarrow A\)  be a surjection in
\(\mathbf {Art}_{\mathbb {K}} \) , we prove that the map
\(Z^1_{\rm sc} (\exp \mathfrak {g}^{\Delta }) (B) \longrightarrow H^1_{\rm sc}(\exp \mathfrak {g}^\Delta ) (B)\times _{H^1_{\rm sc}(\exp \mathfrak {g}^\Delta )(A)}Z^1_{\rm sc} (\exp \mathfrak {g}^{\Delta }) (A),\) 
induced by
\({Z^1_{\rm sc} (\exp \mathfrak {g}^{\Delta })(B) [r]^\beta [d]_\pi & Z^1_{\rm sc} (\exp \mathfrak {g}^{\Delta })(A) [d]^\pi \\H^1_{\rm sc}(\exp \mathfrak {g}^\Delta )(B) [r]_\beta & H^1_{\rm sc}(\exp \mathfrak {g}^\Delta ) (A), \\ }\)
is surjective.
Let \(( [(l,m)],(l_0,m_0)) \in H^1_{\rm sc}(\exp \mathfrak {g}^\Delta ) (B)\times _{H^1_{\rm sc}(\exp \mathfrak {g}^\Delta )(A)}Z^1_{\rm sc} (\exp \mathfrak {g}^{\Delta }) (A)\) , then
\((\beta l,\beta m)\)  and \((l_0,m_0)\)  are gauge equivalent in
\( Z^1_{\rm sc} (\exp \mathfrak {g}^{\Delta })(A)\) , i.e., there exist \(a \in \mathfrak {g}^0_0\otimes \mathfrak {m}_A\)  such that
\(e^a * \beta l = l_0\)  and \(- \beta m \bullet -\partial _{1,1}a \bullet m_0\bullet \partial _{0,1}a=db+[\partial _{0,1}\beta l,b] \) , for some \(b\in {\mathfrak {g}}_1^{-1}\otimes \mathfrak {m}_A\) .
Let \(\tilde{a} \in \mathfrak {g}^0_0 \otimes m_B\)  and \(\tilde{b} \in {\mathfrak {g}}_1^{-1}\otimes \mathfrak {m}_B\)  be liftings of \(a\)  and \(b\) , respectively. The element \((e^{\tilde{a} }*l, \partial _{1,1} \tilde{a} \bullet m \bullet (d\tilde{b}+[\partial _{0,1}l,\tilde{b}]) \bullet -\partial _{0,1} \tilde{a}) \in Z^1_{\rm sc} (\exp \mathfrak {g}^{\Delta })(B)\)  is a pre-image of \(( [(l,m)],(l_0,m_0))\) .
Next, let \(\mathcal {L}\)  be a sheaf of DGLAs on a topological space \(X\) 
and \(\mathcal {U}=\lbrace U_i \rbrace _{i\in I}\)  an open cover. Considering the Čech
cosimplicial DGLA \({\mathcal {L}}({\mathcal {U}})\) , we can define the
functor \(H^1_{\rm sc}(\exp {\mathcal {L}}(\mathcal {U}))\) . This functor depends
on the cover \(\mathcal {U}\) , but as shown in the following Lemma, the limit
over open covers is a well defined functor:
\( H_{\rm Ho}^1(X; \exp \mathcal {L})= \lim _{\begin{array}{c}\longrightarrow \\ \mathcal {U}\end{array}}H^1_{\rm sc}(\exp \mathcal {L}(\mathcal {U})): \mathbf {Art}\rightarrow \mathbf {Set}. \) 
Lemma 2.7 
Let \(\mathcal {U}=\lbrace U_\alpha \rbrace _{\alpha \in I}\)  and \(\mathcal {U}^{\prime }=\lbrace U^{\prime }_\alpha \rbrace _{\alpha \in I^{\prime }}\)  be open covers of \(X\)  with \(\mathcal {U}^{\prime }\)  refinement of
\(\mathcal {U}\)  and let \(\phi , \psi : I^{\prime }\rightarrow I\)  two refinement maps. Then, the
induced morphisms \(\rho _{\phi },\rho _{\psi }:H^1_{\rm sc}(\exp \mathcal {L}(\mathcal {U})) \rightarrow H^1_{\rm sc} (\exp \mathcal {L}(\mathcal {U}^{\prime }))\)  coincide.

Both \(\phi \)  and \(\psi \)  induce, for all \(A\in \mathbf {Art}_{\mathbb {K}}\) , a
morphism \(Z^1_{\rm sc}(\exp \mathcal {L}(\mathcal {U}))(A) \rightarrow Z^1_{\rm sc}(\exp \mathcal {L}(\mathcal {U}^{\prime }))(A)\) , defined sending \((l_i,m_{ij})\)  to
\(\rho _{\phi }(l_i,m_{ij})= ({l_{\phi \alpha }}|_{U^{\prime }_\alpha },{m_{\phi \alpha , \phi \beta }}|_{U^{\prime }_{\alpha \beta }})\)  and \(\rho _{\psi }(l_i,m_{ij})= ({l_{\psi \alpha }}|_{U^{\prime }_\alpha },{m_{\psi \alpha , \psi \beta }}|_{U^{\prime }_{\alpha \beta }})\) ,
respectively.
Therefore, it remains to prove that \(\rho _{\phi }(l_i,m_{ij}) \sim \rho _{\psi }(l_i,m_{ij}) \) , for all \((l_i,m_{ij}) \in Z^1_{\rm sc}(\exp \mathcal {L}(\mathcal {U})) (A)\) , i.e., for all \(\alpha \in I^{\prime }\) , there
exists \(a_\alpha \in \mathcal {L}^0 (U^{\prime }_{\alpha })\otimes m_A\)  such that
\({\left\lbrace \begin{array}{ll}e^{a_\alpha } * {l_{\phi \alpha }}|_{U^{\prime }_\alpha }={l_{\psi \alpha }} |_{U^{\prime }_\alpha } \\- {m_{\phi \alpha , \phi \beta }}|_{U^{\prime }_{\alpha \beta }} \bullet -{a_\alpha } |_{U^{\prime }_{\alpha \beta }}\bullet {m_{\psi \alpha , \psi \beta }}|_{U^{\prime }_{\alpha \beta }} \bullet {a_\beta }|_{U^{\prime }_{\alpha \beta }} \in {\rm Stab}(l_{\phi \beta }|_{U^{\prime }_{\alpha \beta }}).\end{array}\right.}\) 
A simple computation shows that it is enough to choose
\({a_\alpha }:= {m_{\psi \alpha , \phi \alpha }}|_{U^{\prime }_{\alpha }}\) , for all \(\alpha \)  in \(I^{\prime }\) .
Remark 2.8 Having introduced the limit \(H^1_{\rm Ho}(\exp \mathcal {L})\) , for a
sheaf of DGLAs \(\mathcal {L}\)  on a topological space \(X\) , the
results of Section  can be restated as follows:
the functor of infinitesimal deformations of a coherent sheaf
\({\mathcal {F}}\)  on a projective manifold \(X\)  is
\( \operatorname{Def}_{\mathcal {F}}\cong H^1_{\rm Ho}(X;\exp {\mathcal {E}}nd^*(\mathcal {E}^\cdot )),\) 
where \(\mathcal {E}^\cdot \)  is a locally free resolution of \({\mathcal {F}}\) .

The example
of coherent sheaves on projective manifolds together with the DGLA
approach to deformation theory suggests that the functors of Artin
rings \(H^1_{\rm sc}(\exp \mathfrak {g}^\Delta )\)  could actually be isomorphic
to functors \(\operatorname{Def}_{L(\mathfrak {g}^\Delta )}\)  for some DGLA \(L(\mathfrak {g}^\Delta )\) 
canonically associated with \(\mathfrak {g}^\Delta \) . We are going to show
that, under the cohomological hypothesis \(H^{-1}(\mathfrak {g}_2)=0\) , it is
indeed so. More precisely, we are going to prove that, if
\(H^{-1}(\mathfrak {g}_2)=0\) , then the functor of Artin rings \(H^1_{\rm sc}(\exp \mathfrak {g}^\Delta )\)  is isomorphic to the deformation functor
associated with the Thom-Whitney DGLA of the truncation
\({\mathfrak {g}}^{\Delta _{[0,2]}}\) .

The Thom-Whitney DGLA \(\operatorname{Tot}_{TW}({\mathfrak {g}}^\Delta )\)
Let \({\mathfrak {g}}^\Delta \)  be a semicosimplicial DGLA. The maps
\(\partial _i=\partial _{0,i}-\partial _{1,i}+\cdots +(-1)^{i}\partial _{i,i}\) 
endow the vector space \(\bigoplus _i{\mathfrak {g}}_i\)  with the
structure of a differential complex. Moreover, being a
DGLA, each \({\mathfrak {g}}_i\)  is in particular a differential
complex
\({\mathfrak {g}}_i=\bigoplus _j {\mathfrak {g}}_i^j; \qquad d_i\colon {\mathfrak {g}}_i^j\rightarrow {\mathfrak {g}}_i^{j+1}\) 
and since the maps \(\partial _{k,i}\)  are morphisms of DGLAs,
the space
\({\mathfrak {g}}^\bullet _\bullet =\bigoplus _{i,j}{\mathfrak {g}}_i^j\) 
has a natural bicomplex structure. The associated total complex
\(({\rm Tot}({\mathfrak {g}}^\Delta ),d_{\operatorname{Tot}})\quad \text{where}\quad {\rm Tot}({\mathfrak {g}}^\Delta )=\bigoplus _{i}{\mathfrak {g}}_i[-i],\quad d_{\operatorname{Tot}}=\sum _{i,j}\partial _i+(-1)^jd_j\) 
has no
natural DGLA structure. Yet there is an other bicomplex naturally
associated with a semicosimplicial DGLA, whose total complex is naturally a
DGLA.
For every \(n\ge 0\) , denote by \(\Omega _n\)  the differential graded
commutative algebra of polynomial differential forms on the
standard \(n\) -simplex \(\Delta ^n\) :
\( \Omega _n=\frac{{\mathbb {K}}[t_0,\ldots ,t_n,dt_0,\ldots ,dt_n]}{(\sum t_i-1,\sum dt_i)}.\) 
Denote
by \(\delta ^{k,n}\colon \Omega _n\rightarrow \Omega _{n-1}\) , \(k=0,\ldots ,n\) ,
the face maps; then, one has natural morphisms of bigraded DGLAs
\( \delta ^{k,n}\colon \Omega _n\otimes \mathfrak {g}_n\rightarrow \Omega _{n-1}\otimes \mathfrak {g}_n,\qquad \partial _{k,n}\colon \Omega _{n-1}\otimes \mathfrak {g}_{n-1}\rightarrow \Omega _{n-1}\otimes \mathfrak {g}_{n},\) 
for every \(0\le k\le n\) .
The Thom-Whitney bicomplex
is defined as
\(C^{i,j}_{TW}(\mathfrak {g}^\Delta )=\lbrace  (x_n)_{n\in {\mathbb {N}}}\in \bigoplus _n \Omega _n^i\otimes {\mathfrak {g}}_n^j\mid \delta ^{k,n}x_n=\partial _{k,n}x_{n-1}\quad \forall \; 0\le k\le n\rbrace ,\) 
where \(\Omega _n^i\)  denotes the degree \(i\)  component of \(\Omega _n\) .
Its total complex is a DGLA, called the Thom-Whitney DGLA,
and it is denoted by
\(\operatorname{Tot}_{TW}(\mathfrak {g}^\Delta )\) ; denote by
\(d_{TW}\)  the differential of the Thom-Whitney DGLA.
It is a remarkable fact that the integration maps
\( \int _{\Delta ^n}\otimes \operatorname{Id}\colon \Omega _n\otimes \mathfrak {g}_n\rightarrow \mathbb {K}[n]\otimes \mathfrak {g}_n=\mathfrak {g}_n[n]\) 
give a quasi-isomorphism of differential complexes
\(I\colon (\operatorname{Tot}_{TW}({\mathfrak {g}}^\Delta ), d_{TW})\rightarrow (\operatorname{Tot}({\mathfrak {g}}^\Delta ),d_{\rm Tot}).\) 
Moreover, Dupont has described in [8]}, [9]} an
explicit morphism of differential complexes
\(E\colon \operatorname{Tot}({\mathfrak {g}}^\Delta )\rightarrow \operatorname{Tot}_{TW}({\mathfrak {g}}^\Delta )\) 
and an explicit homotopy
\(h\colon \operatorname{Tot}_{TW}({\mathfrak {g}}^\Delta )\rightarrow \operatorname{Tot}_{TW}({\mathfrak {g}}^\Delta )[-1]\) 
such that
\(IE={\rm Id}_{{\rm Tot}({\mathfrak {g}}^\Delta )};\qquad EI-{\rm Id}_{{\rm Tot}_{TW}({\mathfrak {g}}^\Delta )}=[h,d_{TW}].\) 
We also refer to the papers [10]}, [11]}, [12]}
for the explicit description of \(E,h\)  and for the proof of the
above identities. Here, we point out that \(E\)  and \(h\)  are defined
in terms of integration over standard simplexes and
multiplication with canonical differential forms: in particular,
the construction of \(\operatorname{Tot}_{TW}({\mathfrak {g}}^\Delta )\) , \(\operatorname{Tot}({\mathfrak {g}}^\Delta )\) , \(I\) , \(E\) 
and \(h\)  is functorial in the category
\(\mathbf {DGLA}^{\Delta _{\operatorname{mon}}}\)  of
semicosimplicial DGLAs.
Recall that with a DGLA \(L\)  there is
a canonically associated deformation functor \(\operatorname{Def}_L\) , defined as the
solutions of Maurer-Cartan equation modulo gauge action (or, equivalently, modulo homotopy equivalence). Moreover, the tangent space to \(\operatorname{Def}_L\)  is \(H^1(L)\)  and obstructions live in \(H^2(L)\) . Thus,
with a semicosimplicial DGLA \(\mathfrak {g}^{\Delta }\)  is also associated the
deformation functor \(\operatorname{Def}_{{\operatorname{Tot}}_{TW}(\mathfrak {g}^{\Delta })}\) ; its tangent space is
\(T\operatorname{Def}_{{\operatorname{Tot}}_{TW}(\mathfrak {g}^{\Delta })}\cong H^1({\operatorname{Tot}}_{TW}(\mathfrak {g}^{\Delta }))\cong H^1({\operatorname{Tot}}(\mathfrak {g}^{\Delta }))\) 
and obstructions live in
\(H^2({\operatorname{Tot}}_{TW}(\mathfrak {g}^{\Delta }))\cong H^2({\operatorname{Tot}}(\mathfrak {g}^{\Delta })).\) 
Let \(\mathbf {\Delta }^+_{\operatorname{mon}} \)  the
category obtained by adding the empty set \(\emptyset \)  to the
category \(\mathbf {\Delta }_{\operatorname{mon}}\) . An
augmented semicosimplicial differential graded Lie algebra
is a covariant functor \(\mathbf {\Delta }^+_{\operatorname{mon}}\rightarrow \mathbf {DGLA}\) , from the category
\(\mathbf {\Delta }^+_{\operatorname{mon}}\)  to the category of DGLAs.
Equivalently, an augmented semicosimplicial DGLA \({\mathfrak {g}}^{\Delta ^+}\)  is a diagram
\({{{\mathfrak {g}}_{-1}}[r] &{{\mathfrak {g}}_0}@<2pt>[r]@<-2pt>[r] & { {\mathfrak {g}}_1}@<4pt>[r] [r] @<-4pt>[r] & { {\mathfrak {g}}_2}@<6pt>[r] @<2pt>[r] @<-2pt>[r] @<-6pt>[r]& \cdots },\) 
where the truncated diagram \({\mathfrak {g}}^\Delta \)
\({{{\mathfrak {g}}_0} @<2pt>[r]@<-2pt>[r] & {{\mathfrak {g}}_1}@<4pt>[r] [r] @<-4pt>[r] & { {\mathfrak {g}}_2}@<6pt>[r] @<2pt>[r] @<-2pt>[r] @<-6pt>[r]& \cdots }\) 
is a semicosimplicial DGLA and
\(\partial _{0,0}\colon \mathfrak {g}_{-1}\rightarrow \mathfrak {g}_{0}\) 
is a DGLA morphism such that
\(\partial _{0,1}\partial _{0,0}=\partial _{1,1}\partial _{0,0}\) .
Remark 3.1 
There is a morphism of DGLAs
\({\mathfrak {g}}_{-1} &\rightarrow \operatorname{Tot}_{TW}({\mathfrak {g}}^\Delta )\\x &\mapsto (\partial _{0,0}x,\ \partial _{1,1}\partial _{0,0}x,\ \partial _{2,2}\partial _{1,1}\partial _{0,0}x,\dots );\) 

the image of \(x\)  is an element in
\(\operatorname{Tot}_{TW}({\mathfrak {g}}^\Delta )\)  because of equations
\(\partial _{1,1}\partial _{0,0}=\partial _{0,1}\partial _{0,0}\)  and
\(\partial _{k+1,i+1}\partial _{l,i}=\partial _{l,i+1}\partial _{k,i}\) ,
for any \(k\ge l\) . This morphism is obtained as the composition of
the natural inclusion \( {\mathfrak {g}}_{-1} \hookrightarrow \operatorname{Tot}({\mathfrak {g}}^\Delta )\)  with the morphism \(E: \operatorname{Tot}({\mathfrak {g}}^\Delta ) \rightarrow \operatorname{Tot}_{TW}({\mathfrak {g}}^\Delta )\) . The existence of the DGLA
morphism \({\mathfrak {g}}_{-1}\rightarrow \operatorname{Tot}_{TW}({\mathfrak {g}}^\Delta )\) 
is not surprising; indeed, it is induced by the natural morphism
\(\lim {\mathfrak {g}}^\Delta \rightarrow \mathop {\rm holim}{\mathfrak {g}}^\Delta \) .

We use augmentation to link the Thom-Whitney DGLA of the Čech semicosimplicial DGLA of a sheaf of DGLAs with the DGLA of global sections of an acyclic resolution of the sheaf. This result is a translation of Theorem 7.2 in [7]} in terms of the Thom-Whitney DGLA.
We recall that if \({\mathcal {L}}\)  is a sheaf of DGLAs on a topological
space \(X\)  and \({\mathcal {U}}\)  is an open cover of \(X\) , the associated Čech
semicosimplicial differential graded Lie algebra is:
\( \mathcal {L}(\mathcal {U}):\quad { {\prod _i\mathcal {L}(U_i)}@<2pt>[r]@<-2pt>[r] & {\prod _{i<j}\mathcal {L}(U_{ij})}@<4pt>[r] [r] @<-4pt>[r] &{\prod _{i<j<k}\mathcal {L}(U_{ijk})}@<6pt>[r] @<2pt>[r] @<-2pt>[r] @<-6pt>[r]& \cdots }.\) 
A morphism
\(\varphi \colon {\mathcal {L}}\rightarrow {\mathcal {A}}\)  of sheaves of DGLAs is
a quasi-isomorphism if it is a quasi-isomorphism of sheaves of
differential complexes, i.e., if it induces linear isomorphisms
between the cohomology sheaves,
\({\mathcal {H}}^*(\varphi )\colon {\mathcal {H}}^*({\mathcal {L}})\xrightarrow{} {\mathcal {H}}^*({\mathcal {A}}).\) 
Moreover, if \({\mathcal {A}}^k\)  is an acyclic sheaf for any \(k\) , then \(\varphi \colon {\mathcal {L}}\rightarrow {\mathcal {A}} \)  is called an acyclic resolution of \({\mathcal {L}}\) .
Theorem 3.2 
Let \(X\)  be a paracompact Hausdorff topological space, \(\mathcal {L}\) 
a sheaf of differential graded Lie algebras on \(X\) , and
\(\varphi \colon {\mathcal {L}}\rightarrow {\mathcal {A}}\)  an acyclic
resolution. Also let \(A={\mathcal {A}}(X)\)  be the DGLA of global
sections of \({\mathcal {A}}\) . Then, if \(\mathcal {U}\)  is an open
cover of \(X\)  which is acyclic with respect to both \({\mathcal {L}}\)  and \({\mathcal {A}}\) , the DGLA
\({\operatorname{Tot}_{TW}}({\mathcal {L}}(\mathcal {U}))\)  is naturally
quasi-isomorphic to the DGLA \(A\) .

The natural inclusion \(A\rightarrow \mathcal {A}(\mathcal {U})\)  gives an
augmented semicosimplicial DGLA, and so it induces a morphism of DGLAs \(A\rightarrow \operatorname{Tot}_{TW}(\mathcal {A}(\mathcal {U}))\) , that is the composition of the natural inclusion \(A\rightarrow \operatorname{Tot}({\mathcal {A}}({\mathcal {U}}))\)  with the quasi-isomorphism \(E:\operatorname{Tot}({\mathcal {A}}({\mathcal {U}}))\rightarrow \operatorname{Tot}_{TW}(\mathcal {A}(\mathcal {U}))\) , by Remark REF .
Since the sheaves \({\mathcal {A}}^k\)  are acyclic and
\({\mathcal {U}}\) -acyclic, and \(A^k=H^0(X;\mathcal {A}^k)\) , the inclusion \(A \rightarrow \operatorname{Tot}(\mathcal {A}(\mathcal {U}))\)  is a
quasiisomorphism. Indeed, we have a natural identification
\(H^*({\operatorname{Tot}}(\mathcal {A}(\mathcal {U})))={\mathbb {H}}^*(X;{\mathcal {A}})\) , and the spectral sequence abutting to the
hypercohomology of \(X\)  with coefficients in \({\mathcal {A}}\) 
degenerates at \(E_2\) , giving
\({\mathbb {H}}^k(X;{\mathcal {A}})=\bigoplus _{p+q=k}E_2^{p,q}=E_2^{k,0}=H^k(A).\) 
Then, \(A\rightarrow \operatorname{Tot}_{TW}(\mathcal {A}(\mathcal {U}))\)  is a quasi-isomorphism of DGLAs.
The morphism \(\varphi \colon {\mathcal {L}}\rightarrow {\mathcal {A}}\)  induces a
morphism of semicosimplicial DGLAs
\( \varphi \colon \mathcal {L}(\mathcal {U})\rightarrow \mathcal {A}(\mathcal {U}), \) 
and a morphism of complexes
\( \varphi \colon \operatorname{Tot}_{TW}(\mathcal {L}(\mathcal {U})) \rightarrow \operatorname{Tot}_{TW}(\mathcal {A}(\mathcal {U})).\) 
Since the open
cover \({\mathcal {U}}\)  is \({\mathcal {L}}\) -acyclic, the cohomology
of the total complex \(\operatorname{Tot}(\mathcal {L}(\mathcal {U}))\)  is naturally
identified with the hypercohomology of \(X\)  with coefficients in
\({\mathcal {L}}\) ,
\(H^*(\operatorname{Tot}(\mathcal {L}(\mathcal {U})))\cong {\mathbb {H}}^*(X;{\mathcal {L}}),\) 
and the induced linear map
\(H^*(\varphi )\colon H^*(\operatorname{Tot}(\mathcal {L}(\mathcal {U})))\rightarrow H^*(\operatorname{Tot}(\mathcal {A}(\mathcal {U})))\) 
is identified with the linear map
\({\mathbb {H}}^*(\varphi )\colon {\mathbb {H}}^*(X;{\mathcal {L}})\rightarrow {\mathbb {H}}^*(X;{\mathcal {A}})\) 
induced in hypercohomology. Since, by hypothesis, \(\varphi \)  is a
quasi-isomorphism of sheaves of DGLAs, the induced map in
hypercohomology is an isomorphism, and so the morphism
\(\varphi \colon \operatorname{Tot}(\mathcal {L}(\mathcal {U})) \rightarrow \operatorname{Tot}(\mathcal {A}(\mathcal {U}))\)  is a
quasi-isomorphism of complexes.
Via the composition with quasi-isomorphisms \(E\)  and \(I\)  between the total complex and the Thom-Whitney total complex of a semicosimplicial DGLA, the morphism \(\varphi \)  induces a quasi-isomorphism of DGLAs
\( \operatorname{Tot}_{TW}(\mathcal {L}(\mathcal {U})) \rightarrow \operatorname{Tot}_{TW}(\mathcal {A}(\mathcal {U})).\) 
Therefore, we have the
chain of quasi-isomorphisms of DGLAs
\(\operatorname{Tot}_{TW}(\mathcal {L}(\mathcal {U}))\xrightarrow{}\operatorname{Tot}_{TW}(\mathcal {A}(\mathcal {U}))\xleftarrow{}A.\) 

Truncations
Let
\( \mathfrak {g}^{\Delta }: \ \ { \mathfrak {g}_0@<2pt>[r]@<-2pt>[r] & \mathfrak {g}_1@<4pt>[r] [r] @<-4pt>[r] & \mathfrak {g}_2@<6pt>[r] @<2pt>[r] @<-2pt>[r] @<-6pt>[r] &\mathfrak {g}_3 @<8pt>[r] @<4pt>[r] [r]@<-4pt>[r] @<-8pt>[r] &\ldots }\) 
be a semicosimplicial DGLA. Let \(m_1 \in \mathbb {N}\)  and \(m_2 \in \mathbb {N}\cup \lbrace \infty \rbrace \)  with \(m_1 \le m_2\) , we denote by
\(\mathfrak {g}^{\Delta _{[m_1,m_2]}}\)  the truncated between levels \(m_1\) 
and \(m_2\) semicosimplicial DGLA defined by
\((\mathfrak {g}^{\Delta _{[m_1,m_2]}})_n={\left\lbrace \begin{array}{ll}\mathfrak {g}_n &\text{for }m_1 \le n\le m_2\\0&\text{otherwise},\end{array}\right.}\) 
with the obvious maps
\(\partial _{k,i}^{[m_1,m_2]}=\partial _{k,i}\) , for \(m_1 < i\le m_2\) ,
and \(\partial _{k,i}^{[m_1,m_2]}=0\) , otherwise. For any positive
integers \(m_1,m_2,r_1,r_2\) , such that \(r_i\le m_i\) , the map
\(\operatorname{Id}_{[m_1,r_2]}\colon \mathfrak {g}^{\Delta _{[m_1,m_2]}}\rightarrow \mathfrak {g}^{\Delta _{[r_1,r_2]}}\)  given by
\(\operatorname{Id}_{[m_1,r_2]}\biggr \vert _{(\mathfrak {g}^{\Delta _{[m_1,m_2]}})_n}={\left\lbrace \begin{array}{ll}\operatorname{Id}_{\mathfrak {g}_n}&\text{if } m_1 \le n \le r_2\\0&\text{otherwise}.\end{array}\right.}\) 
is a morphism of semicosimplicial DGLAs; it induces the natural
morphism of complexes \(\phi : \operatorname{Tot}(\mathfrak {g}^{\Delta _{[m_1,m_2]}})\rightarrow \operatorname{Tot}(\mathfrak {g}^{\Delta _{[r_1,r_2]}})\)  and the natural morphism of DGLAs
\(\psi : \operatorname{Tot}_{TW}(\mathfrak {g}^{\Delta _{[m_1,m_2]}})\rightarrow \operatorname{Tot}_{TW}(\mathfrak {g}^{\Delta _{[r_1,r_2]}})\) . Note that we have an homotopy
commutative diagram of complexes
\( { \operatorname{Tot}(\mathfrak {g}^{\Delta _{[m_1,m_2]}}) @<1ex>[d]^{E}[r]^{\phi }& \operatorname{Tot}(\mathfrak {g}^{\Delta _{[r_1,r_2]}}) @<1ex>[d]^{E} \\\operatorname{Tot}_{TW}(\mathfrak {g}^{\Delta _{[m_1,m_2]}}) [r]^{\psi }@<1ex>[u]^{I}& \operatorname{Tot}_{TW}(\mathfrak {g}^{\Delta _{[r_1,r_2]}})@<1ex>[u]^{I}.} \) 
Proposition 4.1 
Let \(\mathfrak {g}^{\Delta }\)  be a semicosimplicial DGLA such that
\(H^j(\mathfrak {g}_i)=0\) , for all \(i\ge 0\)  and \(j<0\) . Then, the morphism
\(\operatorname{Id}_{[0,2]}\)  induces a natural isomorphism of functors:
\(\operatorname{Def}_{\operatorname{Tot}_{TW}(\mathfrak {g}^\Delta ) } \xrightarrow{}\operatorname{Def}_{\operatorname{Tot}_{TW}(\mathfrak {g}^{\Delta _{[0,2]}})}.\) 

It is a well known fact (see, e.g., [14]} for a
proof), that a DGLA morphism which is surjective on \(H^0\) , bijective on \(H^1\)  and injective on \(H^2\)  induces an
isomorphism between the associated deformation functors. Since the above
homotopy commutative diagram identifies \(H^*(\psi )\)  with
\(H^*(\phi )\) , it is enough to prove that \(H^0(\phi )\)  is
surjective, \(H^1(\phi )\)  is bijective and \(H^2(\phi )\)  is injective. This
is easily checked by looking at the spectral sequences
associated with double complexes of \({\mathfrak {g}}^\Delta \)  and \({\mathfrak {g}}^{\Delta _{[0,2]}}\) .
Remark 4.2 
Observe that, for any semicosimplicial DGLA \(\mathfrak {g}^{\Delta }\) , we have
\(Z^1_{\rm sc}(\exp \mathfrak {g}^\Delta )= Z_{\rm sc}^1 (\exp \mathfrak {g}^{\Delta _{[0,2]}})\)  and \(H^1_{\rm sc}(\exp \mathfrak {g}^\Delta )= H^1_{\rm sc}(\exp \mathfrak {g}^{\Delta _{[0,2]}})\) . Moreover, the inclusion \(Z^1_{\rm sc}(\exp \mathfrak {g}^\Delta )\hookrightarrow Z^1_{\rm sc} (\exp \mathfrak {g}^{\Delta _{[0,1]}})\)  induces an injective
map \(H^1_{\rm sc}(\exp \mathfrak {g}^\Delta ) \hookrightarrow H^1_{\rm sc}(\exp \mathfrak {g}^{\Delta _{[0,1]}})\) .

Remark 4.3 
For later use, we point out that, if \(\mathfrak {g}^{\Delta }\)  is a semicosimplicial DGLA with \(H^{-1}(\mathfrak {g}_2)=0\) , then
\(\operatorname{Def}_{\operatorname{Tot}_{TW}(\mathfrak {g}^{\Delta _{[2,2]}})}\)  is trivial. Indeed, \(H^{1}(\operatorname{Tot}_{TW}( \mathfrak {g}^{ \Delta _{[2,2]}}) )= H^{-1}(\mathfrak {g}_2)=0\) .

Remark 4.4 Note that, by the definition of \(H^1_{\rm sc}(\exp \mathfrak {g}^\Delta )\)  it follows that, if \(H^{-1}(\mathfrak {g}_2)=0\) , then
\(TH^1_{\rm sc}(\exp \mathfrak {g}^\Delta )=H^1({\operatorname{Tot}}(\mathfrak {g}^{\Delta _{[0,2]}})).\) 
Hence, the two functors of Artin rings \(H^1_{\rm sc}(\exp \mathfrak {g}^\Delta )\)  and \(\operatorname{Def}_{{\operatorname{Tot}}_{TW}(\mathfrak {g}^{\Delta _{[0,2]}})}\)  have naturally isomorphic tangent spaces when \(H^{-1}(\mathfrak {g}_2)=0\) . We will show in Section  that in this case these two functors are actually isomorphic.


A lemma on Maurer-Cartan elements
We will now give an explicit description of the solutions of Maurer-Cartan equation for the DGLAs
\(\operatorname{Tot}_{TW}(\mathfrak {g}^{\Delta _{[0,1]}})\)  and \(\operatorname{Tot}_{TW}(\mathfrak {g}^{\Delta _{[0,2]}})\) . Our main tool will be the following general result [15]}:
Lemma 5.1 
Let \((L,d,[~,~])\)  be a differential graded Lie algebra such that:

\(L=M\oplus C\oplus D\)  as graded vector spaces.

\(M\)  is a differential graded subalgebra of \(L\) .

\(d\colon C\rightarrow D[1]\)  is an isomorphism of graded vector
spaces.

Then, for every \(A\in \mathbf {Art}_{\mathbb {K}}\)  there exists a bijection
\( \alpha \colon \operatorname{MC}_M(A)\times (C^0\otimes \mathfrak {m}_A){\stackrel{\sim }{\longrightarrow }}\operatorname{MC}_L(A),\qquad (x,c)\mapsto e^c\ast x.\) 

As almost immediate corollaries we obtain:
Proposition 5.2 
Let \(\mathfrak {g}^{\Delta }\)  be a semicosimplicial DGLA. Then, for every \(A\in \mathbf {Art}_{\mathbb {K}}\) , the solutions of the Maurer-Cartan equation for the Thom-Whitney DGLA
\(\operatorname{Tot}_{TW}(\mathfrak {g}^{\Delta _{[0,1]}})\otimes \mathfrak {m}_A\)  are of the form
\((x, e^{p(t)}* \partial _{0,1}x)\) ,
where \(x \in \operatorname{MC}_{\mathfrak {g}_0}(A)\)  and \(p(t) \in (\mathfrak {g}_1^0[t]\cdot t) \otimes \mathfrak {m}_A\) . The elements \(x, p\)  are uniquely determined, and they satisfy
\(\partial _{1,1}x=e^{p(1)}*\partial _{0,1}x.\) 

Notice that \(\operatorname{Tot}_{TW}(\mathfrak {g}^{\Delta _{[0,1]}})\)  is a sub-DGLA of
\(\mathfrak {g}_0\oplus \Omega _1\otimes \mathfrak {g}_1\) . Then, Lemma REF 
with the decomposition of \(\Omega _1\otimes \mathfrak {g}_1\)  given by
\( M=\mathfrak {g}_1, \qquad C= \mathfrak {g}_1[t]\cdot t, \qquad D=dC \) 
tells us that every solution of the
Maurer-Cartan equation for \(\operatorname{Tot}_{TW}(\mathfrak {g}^{\Delta _{[0,1]}})\otimes \mathfrak {m}_A\)  is of the form specified above.
Proposition 5.3 Let \(\mathfrak {g}^{\Delta }\)  be a semicosimplicial DGLA. Then, for every
\(A\in \mathbf {Art}_{\mathbb {K}}\) , the solutions of the Maurer-Cartan
equation for the Thom-Whitney DGLA
\(\operatorname{Tot}_{TW}(\mathfrak {g}^{\Delta _{[0,2]}})\otimes \mathfrak {m}_A\)  are of the form
\((x, e^{p(t)}* \partial _{0,1}x, e^{q(s_0,s_1)+ r(s_0,s_1,ds_0,ds_1)}*\partial _{0,2}\partial _{0,1}x),\) 
where \(x \in \operatorname{MC}_{\mathfrak {g}_0}(A)\) , \(p(t) \in (\mathfrak {g}_1^0[t]\cdot t)\otimes \mathfrak {m}_A\) , \(q(s_0,s_1) \in (\mathfrak {g}_2^0[s_0,s_1]\cdot s_0+ \mathfrak {g}_2^0[s_0,s_1]\cdot s_1)\otimes \mathfrak {m}_A\)  and \(r(s_0,s_1,ds_0,ds_1) \in (\mathfrak {g}_2^{-1}[s_0,s_1]\cdot s_0ds_1) \otimes \mathfrak {m}_A\) . The elements \(x,p,q,r\)  are uniquely determined, and they satisfy
\({\left\lbrace \begin{array}{ll}\partial _{1,1}x=e^{p(1)}*\partial _{0,1}x, \\\partial _{0,2}p(t)= q(0,t), \\\partial _{1,2}p(t)= q(t,0), \\e^{(- \partial _{2,2}p(t))\bullet (q(t,1-t)+ r(t,1-t,dt))\bullet (-q(0,1))}*\partial _{2,2}\partial _{0,1}x=\partial _{2,2}\partial _{0,1}x.\end{array}\right.}\) 

Since \(\operatorname{Tot}_{TW}(\mathfrak {g}^{\Delta _{[0,2]}})\)  is a sub-DGLA of
\(\mathfrak {g}_0\oplus \Omega _1\otimes \mathfrak {g}_1\oplus \Omega _2\otimes \mathfrak {g}_2\) ,
applying Lemma REF 
with the decomposition of \(\Omega _2\otimes \mathfrak {g}_2\)  given by
\( M=\mathfrak {g}_2,\qquad C=\mathfrak {g}_2[s_0,s_1]\cdot s_0+ \mathfrak {g}_2[s_0,s_1]\cdot s_1 +\mathfrak {g}_2[s_0,s_1]\cdot s_0ds_1 ,\qquad D=dC \) 
we obtain that every solution of the
Maurer-Cartan equation for \(\operatorname{Tot}_{TW}(\mathfrak {g}^{\Delta _{[0,2]}})\otimes \mathfrak {m}_A\)  is of
the form
\((x, e^{p(t)}* y, e^{q(s_0,s_1)+ r(s_0,s_1,ds_0,ds_1)}*z),\) 
with the face conditions
\(y=\partial _{0,1}x;\qquad z=\partial _{0,2}\partial _{0,1}x.\) 
The first relations in (REF ) are a direct consequence
of face conditions and uniqueness. The last one is obtained as
follows. The last face condition is
\( \partial _{2,2}(e^{p(t)}* \partial _{0,1}x) = e^{q(t,1-t)+ r(t,1-t,dt)}*\partial _{0,2}\partial _{0,1}x;\) 
using the other face conditions and relations
between maps \(\partial _{k,i}\) , we obtain that
\(\partial _{2,2} \partial _{0,1}x =\partial _{0,2}\partial _{1,1} x = \partial _{0,2}(e^{p(1)}* \partial _{0,1}x )= e^{q(0,1)}*\partial _{0,2}\partial _{0,1}x.\) 
Then, the above equation
becomes
\( e^{\partial _{2,2}p(t)}* \partial _{2,2}\partial _{0,1}x =e^{(q(t,1-t)+ r(t,1-t,dt))\bullet (-q(0,1))}* \partial _{2,2}\partial _{0,1}x. \) 

The isomorphism \(H^1_{\rm sc}(\exp \mathfrak {g}^{\Delta _{[0,1]}}) \cong \operatorname{Def}_{ \operatorname{Tot}_{TW}(\mathfrak {g}^{ \Delta _{[0,1]} })}\)
Proposition 6.1 
Let \(\mathfrak {g}^\Delta \)  be a semicosimplicial DGLA. The map
\(\Phi _{[0,1]}:\operatorname{MC}_{ \operatorname{Tot}_{TW}(\mathfrak {g}^{\Delta _{[0,1]}}) }(A)\rightarrow (\mathfrak {g}_0^1\oplus \mathfrak {g}_1^0 ) \otimes \mathfrak {m}_A,\) 
given by
\((x, e^{p(t)}* \partial _{0,1}x)\mapsto (x, p(1) ),\) 
induces a natural transformation of functors of Artin rings
\( \operatorname{Def}_{\operatorname{Tot}_{TW}(\mathfrak {g}^{\Delta _{[0,1]}})}\rightarrow H^1_{\rm sc}(\exp \mathfrak {g}^{\Delta _{[0,1]}}).\) 

Clearly, if \((x, e^{p(t)}* \partial _{0,1}x) \in \operatorname{MC}_{ \operatorname{Tot}_{TW}(\mathfrak {g}^{\Delta _{[0,1]}}) }(A)\) , then \((x,p(1))\in Z^1_{\rm sc}(\exp \mathfrak {g}^{\Delta _{[0,1]}})\) .
We have to show that if two elements \(\eta _0=(x_0,e^{p_0(t)}*\partial _{0,1}x_0)\)  and
\(\eta _1=(x_1,e^{p_1(t)}*\partial _{0,1}x_1)\)  in \(\operatorname{MC}_{{\operatorname{Tot}}_{TW}(\mathfrak {g}^{\Delta _{[0,1]}})}(A)\) 
are homotopy equivalent, then \(\Phi _{[0,1]}(\eta _0)\sim \Phi _{[0,1]}(\eta _1)\)  in
\(Z^1_{\rm sc}(\exp \mathfrak {g}^{\Delta _{[0,1]}})\) . Let \(z(\xi ,d\xi )\)  be an homotopy
between \(\eta _0\)  and \(\eta _1\) . Therefore, \(z(\xi ,d\xi )\)  is a Maurer-Cartan element for \({\operatorname{Tot}}_{TW}(\mathfrak {g}^{\Delta _{[0,1]}})[\xi ,d\xi ]\)  and so, reasoning as in the proof of Proposition REF , we find
\(z(\xi ,d\xi )= (e^{T(\xi )}* u, e^{U(t,dt;\xi )}* v),\) 
with \(T(0)=U(t,dt;0)=0\) .
Since \(z(0)=\eta _0\) , we get
\(z(\xi ,d\xi )=(e^{T(\xi )}* x_0, e^{U(t,dt; \xi )}* e^{p_0(t)}*\partial _{0,1}x_0).\) 
The face conditions for \(z(\xi ,d\xi )\)  and uniqueness imply
\( U(0;\xi )=\partial _{0,1}T(\xi ) \quad \mbox{ }\quad U(1;\xi )=\partial _{1,1}T(\xi ). \) 
Moreover, \(z(1)=\eta _1\) , and so
\( (e^{T(1)}* x_0, e^{U(t,dt; 1)}* e^{p_0(t)}* \partial _{0,1}x_0)=(x_1,e^{p_1(t)}*\partial _{0,1}x_1); \) 
by uniqueness again, we have
\( e^{T(1)}*x_0=x_1.\) 
Furthermore, \(e^{U(t,dt; 1)}* e^{p_0(t)}* \partial _{0,1}x_0 = e^{p_1(t)}*\partial _{0,1}x_1\) ,
so, using the face conditions for \(\eta _0\)  and \(\eta _1\) , we obtain
\(\partial _{0,1} x_0 = e^{-p_0(t)\bullet -U(t,dt; 1) \bullet p_1(t) \bullet \partial _{0,1} T(1)} *\partial _{0,1} x_0\) 
Next, we recall [16]} that if \(L\)  is a DGLA, \(x(t,dt)\)  is a Maurer-Cartan element for \(L[t,dt]\)  and \(\mu (t,dt)\in L[t,dt]^0\)  is such that \(e^{\mu (t,dt)}*x(t,dt)=x(t,dt)\) , then \(\mu (1)\)  is an element of the irrelevant stabilizer of \(x(1)\) . Therefore, in our case we get
\(-p_0(1)\bullet -\partial _{1,1}T(1) \bullet p_1(1) \bullet \partial _{0,1} T(1) \in {\rm Stab}(\partial _{0,1} x_0).\) 
Proposition 6.2 
Let \(\mathfrak {g}^{\Delta }\)  be a semicosimplicial DGLA. The map
\(\Phi _{[0,1]}:\operatorname{Def}_{ \operatorname{Tot}_{TW}(\mathfrak {g}^{\Delta _{[0,1]}}) } \rightarrow H^1_{\rm sc}(\exp \mathfrak {g}^{\Delta _{[0,1]}}) \) 
is an isomorphism of functors of Artin rings. In particular, \(H^1_{\rm sc}(\exp \mathfrak {g}^{\Delta _{[0,1]}})\) 
is a deformation functor.

Let \(\Psi _{[0,1]}: Z^1_{\rm sc}(\exp \mathfrak {g}^{\Delta _{[0,1]}})(A) \rightarrow \operatorname{Tot}_{TW}(\mathfrak {g}^{\Delta _{[0,1]}})\otimes {\mathfrak {m}}_A \)  be the map given by \( (l, m) \mapsto (l, e^{t m}* \partial _{0,1}l)\) ; it is immediate to check that \(\Phi _{[0,1]}\)  actually takes its values in \(\operatorname{MC}_{ \operatorname{Tot}_{TW}(\mathfrak {g}^{\Delta _{[0,1]}}) }(A)\) . Moreover, \(\Psi _{[0,1]}\)  induces a map
\(H^1_{\rm sc}(\exp \mathfrak {g}^{\Delta _{[0,1]}})(A)\rightarrow \operatorname{Def}_{ \operatorname{Tot}_{TW}(\mathfrak {g}^{\Delta _{[0,1]}}) }(A),\) 
which is the inverse of \(\Phi _{[0,1]}\) .
Indeed, if \((l_0,m_0)\sim (l_1,m_1)\)  in \(Z^1_{\rm sc}(\exp \mathfrak {g}^{\Delta _{[0,1]}})(A)\) , then
there exist elements \(a \in \mathfrak {g}^0_0\otimes \mathfrak {m}_A\)  and \(b\in {\mathfrak {g}}_1^{-1}\otimes {\mathfrak {m}}_A\)  such that
\({\left\lbrace \begin{array}{ll}e^a * l_0=l_1\\- m_0\bullet -\partial _{1,1}a \bullet m_1\bullet \partial _{0,1}a=db+[\partial _{0,1}l_0,b].\end{array}\right.}\) 
Therefore, the images \((l_0, e^{t m_0}* \partial _{0,1}l_0)\)  and \((l_1, e^{tm_1}* \partial _{0,1}l_1)\)  are homotopic via the element
\( z(\xi , d\xi )= ( e^{\xi a}* l_0, e^{t\bigl (\partial _{1,1}(\xi a)\bullet m_0 \bullet (d(\xi b) +[\partial _{0,1}l_0, \xi b]) \bullet -\partial _{0,1}(\xi a)\bigr )\bullet \partial _{0,1}(\xi a)}* \partial _{0,1} l_0 ).\) 
The composition \(\Phi _{[0,1]}\circ \Psi _{[0,1]}\colon Z^1_{\rm sc}(\exp \mathfrak {g}^{\Delta _{[0,1]}})(A)\rightarrow Z^1_{\rm sc}(\exp \mathfrak {g}^{\Delta _{[0,1]}})(A)\)  is clearly the
identity, whereas the composition
\(\Psi _{[0,1]} \circ \Phi _{[0,1]}: \operatorname{MC}_{\operatorname{Tot}_{TW}(\mathfrak {g}^{\Delta _{[0,1]}}) }(A) \rightarrow \operatorname{MC}_{ \operatorname{Tot}_{TW}(\mathfrak {g}^{\Delta _{[0,1]}})}(A)\)  is homotopic to the identity. Indeed, \((x, e^{p(t)}* \partial _{0,1}x)\)  and
\((x, e^{t p(1)}* \partial _{0,1}x)\)  are homotopic in \(\operatorname{MC}_{\operatorname{Tot}_{TW}(\mathfrak {g}^{\Delta _{[0,1]}}) }(A)\)  via the element
\(z(\xi ,d\xi )= (x, e^{\xi t p(1) + (1-\xi ) p(t) }*\partial _{0,1}x)\) .
Remark 6.3 A particular case of Proposition REF , with an almost identical proof, has been considered by one of the authors in [16]}. Namely, given three DGLAs \(L,M\)  and \(N\)  and two DGLA morphisms \(h\colon L\rightarrow M\)  and \(g\colon N\rightarrow M\) , one can consider the semicosimplicial DGLA
\({ L\oplus N@<2pt>[r]^{(0,g)}@<-2pt>[r]_{(h,0)} & { M}@<4pt>[r] [r] @<-4pt>[r] & 0@<6pt>[r] @<2pt>[r] @<-2pt>[r] @<-6pt>[r]&\cdots }\) 
to reobtain [16]}.


Proof of the main theorem
In this section, we prove the existence of a natural isomorphism
of functors of Artin rings \(H^1_{\rm sc}(\exp \mathfrak {g}^\Delta ) \cong \operatorname{Def}_{ \operatorname{Tot}_{TW}(\mathfrak {g}^{ \Delta _{[0,2]} })}\) , for any
semicosimplicial DGLA \(\mathfrak {g}^{\Delta }\)  such that \(H^{-1}(\mathfrak {g}_2)=0\) .
As an immediate consequence we obtain a natural isomorphism of deformation
functors \(H^1_{\rm sc}(\exp \mathfrak {g}^\Delta ) \cong \operatorname{Def}_{{\operatorname{Tot}_{TW}}(\mathfrak {g}^{ \Delta })}\) , for any semicosimplicial DGLA
\(\mathfrak {g}^{\Delta }\) , such that \(H^{j}(\mathfrak {g}_i)=0\)  for \(i\ge 0\)  and \(j<0\) .
The proof is considerably harder than in the case \(\mathfrak {g}^{\Delta _{[0,1]}}\)  considered in the previous section. Indeed, we are still able to define a map \(\Phi \colon \operatorname{MC}_{\operatorname{Tot}_{TW}(\mathfrak {g}^{\Delta _{[0,2]}})}\rightarrow Z^1_{\rm sc}(\exp \mathfrak {g}^\Delta )\)  inducing a natural transformation \(\operatorname{Def}_{\operatorname{Tot}_{TW}(\mathfrak {g}^{\Delta _{[0,2]}})}\rightarrow H^1_{\rm sc}(\exp \mathfrak {g}^\Delta )\) , but we will not be able to explicitly define an homotopy inverse to \(\Phi \) , so we will have to directly check that the map \(\operatorname{Def}_{\operatorname{Tot}_{TW}(\mathfrak {g}^{\Delta _{[0,2]}})}\rightarrow H^1_{\rm sc}(\exp \mathfrak {g}^\Delta )\)  is an isomorphism.
Proposition 7.1 
Let \(\mathfrak {g}^\Delta \)  be a semicosimplicial DGLA. The map
\(\Phi :\operatorname{MC}_{ \operatorname{Tot}_{TW}(\mathfrak {g}^{\Delta _{[0,2]}}) }(A)\rightarrow (\mathfrak {g}_0^1\oplus \mathfrak {g}_1^0 ) \otimes \mathfrak {m}_A,\) 
given by
\((x, e^{p(t)}* \partial _{0,1}x, e^{q(s_0,s_1)+ r(s_0,s_1,ds_1,ds_1)}*\partial _{0,2}\partial _{0,1}x)\mapsto (x, p(1) ),\) 
induces a natural transformation of functors of Artin rings
\( \operatorname{Def}_{\operatorname{Tot}_{TW}(\mathfrak {g}^{\Delta _{[0,2]}})}\rightarrow H^1_{\rm sc}(\exp \mathfrak {g}^\Delta ).\) 

First we check that \(\Phi \)  takes its values in \(Z^1_{\rm sc}(\exp \mathfrak {g}^\Delta )(A)\) .
The only nontrivial point consists in showing that \(- \partial _{2,2}p(1) \bullet \partial _{1,2}p(1)\bullet -\partial _{0,2}p(1)\)  is an element of the irrelevant stabilizer of \(\partial _{2,2}\partial _{0,1}x\) . This follows by the face condition
\(e^{(- \partial _{2,2}p(t))\bullet (q(t,1-t)+ r(t,1-t,dt))\bullet (-q(0,1))}*\partial _{2,2}\partial _{0,1}x=\partial _{2,2}\partial _{0,1}x,\) 
applying [16]} once again. Next, we notice that
the equivalence relation \(\sim \)  on \(Z^1_{\rm sc}(\exp \mathfrak {g}^\Delta )(A)\)  only involves the DGLAs \({\mathfrak {g}}_0\)  and
\({\mathfrak {g}}_1\) ; hence, we can conclude verbatim following the
proof of Proposition REF .
Proposition 7.2 
The map \(\Phi : \operatorname{Def}_{{\operatorname{Tot}}_{TW}(\mathfrak {g}^{\Delta _{[0,2]}})}(A)\rightarrow H^1_{\rm sc}(\exp \mathfrak {g}^\Delta )(A)\)  is surjective.

Let \((l,m)\in Z^1_{\rm sc}(\exp \mathfrak {g}^\Delta )(A)\)  and \(n\in \mathfrak {g}_2^{-1}\otimes \mathfrak {m}_A\) , such that \( \partial _{0,2}m\bullet - \partial _{1,2}m \bullet \partial _{2,2}m = dn +\frac{1}{2} [\partial _{2,2}\partial _{0,1}l, n]\) .
Consider the element \(w(t) = d(tn) + \frac{1}{2} [\partial _{2,2}\partial _{0,1}l,t n]\)  in the irrelevant stabilizer of \(\partial _{2,2}\partial _{0,1}l\)  and
\( R(s_0,s_1)=s_0s_1\frac{s_0\partial _{2,2}m \bullet -w(s_0) \bullet s_0\partial _{0,2}m \bullet -s_0 \partial _{1,2}m}{s_0 (1-s_0) }\bullet s_0 \partial _{1,2}m \bullet s_1 \partial _{0,2}m. \) 
Then,
\( (l, e^{tm}* \partial _{0,1}l, e^{R(s_0,s_1)}*\partial _{0,2}\partial _{0,1}l) \) 
is an element in \(\operatorname{MC}_{\operatorname{Tot}_{TW}(\mathfrak {g}^{\Delta _{[0,2]}})}(A)\)  in the
fiber of \(\Phi \)  over \((l,m)\) . Indeed, clearly it satisfies the Maurer-Cartan equation in \(\mathfrak {g}_0 \oplus \mathfrak {g}_1\otimes \Omega _1 \oplus \mathfrak {g}_2\otimes \Omega _2\) ; the first face
conditions follow easly noticing that
\(R(0 ,t )= t \partial _{0,2}m\)  and \(R(t,0)= t\partial _{1,2}m\) ;
for the last one, we have:
\( e^{R(t,1-t)}*\partial _{0,2}\partial _{0,1}l =e^{ t\partial _{2,2}m \bullet -w(t) \bullet \partial _{0,2}m} *\partial _{0,2}\partial _{0,1}l=\) 
\(= e^{ t\partial _{2,2}m \bullet -w(t)} *\partial _{0,2}\partial _{1,1}l =e^{ t\partial _{2,2}m \bullet -w(t)} * \partial _{2,2}\partial _{0,1}l = e^{ t\partial _{2,2}m} *\partial _{2,2}\partial _{0,1}l. \) 
We will prove that the map
\(\Phi :\operatorname{Def}_{\operatorname{Tot}_{TW}(\mathfrak {g}^{\Delta _{[0,2]}})} (A) \rightarrow H^1_{\rm sc}(\exp \mathfrak {g}^\Delta )(A)\)  is injective, under the hypothesis \(H^{-1}(\mathfrak {g}_2)=0\) . For this we need two remarks.
Remark 7.3 
Let \((L,d,[\ , \ ])\)  be a DGLA, \(A\in \bf {Art}_{\mathbb {K}}\)  and
\(x\in L^1\otimes \mathfrak {m}_A\) . The linear endomorphism \(d_x=d+[x , \ ]\)  of
\(L\otimes \mathfrak {m}_A\)  is a differential if and only if \(x\in \operatorname{MC}_L(A)\) , and in this case
\((L\otimes \mathfrak {m}_A, d_x, [\ , \ ])\)  is a DGLA. So,
we can define the set of the Maurer-Cartan elements
\(\operatorname{MC}_{L}^{x}(A)\)  and the gauge action of \((L^0\otimes {\mathfrak {m}}_A, d_x, [\ , \ ])\)  on it. We denote by \(\operatorname{Def}_L^x(A)\)  the quotient of
\(\operatorname{MC}_L^x(A)\)  with respect to the gauge action. The affine map
\( \begin{array}{rll}L \otimes \mathfrak {m}_A & \rightarrow & L \otimes \mathfrak {m}_A \\v & \mapsto & v-x.\end{array}\) 
induces an isomorphism \(\operatorname{Def}_L(A)\cong \operatorname{Def}_L^x(A)\)  with obvious inverse \(v\mapsto v+x\) .
Next, let \(M\subseteq L\)  be a sub-DGLA and let \(x \in \operatorname{MC}_L(A)\) . If
\(M\otimes \mathfrak {m}_A\)  is closed under the differential \(d_x\) , then we can consider the set of Maurer-Cartan elements
\(\operatorname{MC}_{M}^{x}(A)\) , and its quotient \(\operatorname{Def}_M^x(A)\) . The tangent space to \(\operatorname{Def}_M^x(A)\)  is \(H^1(M\otimes {\mathfrak {m}}_A,d_x)\) ; so, by upper semicontinuity of cohomology, \(H^1(M,d)=0\)  implies that \(\operatorname{Def}^x_{M}(A)\)  is trivial, for all \(x\in \operatorname{MC}_L(A)\)  such that \(d_x(M\otimes \mathfrak {m}_A)\subseteq M\otimes \mathfrak {m}_A\) .

Remark 7.4 
For any semicosimplicial DGLA \(\mathfrak {g}^{\Delta }\) , the truncation morphism
\(\operatorname{Tot}^0_{TW}(\mathfrak {g}^{\Delta _{[0,2]}}) \rightarrow \operatorname{Tot}^0_{TW}(\mathfrak {g}^{\Delta _{[0,1]}})\) 
is surjective, i.e., for any \((a_0,a_1) \in \operatorname{Tot}^0_{TW}(\mathfrak {g}^{\Delta _{[0,1]}})\)  there exist \(a_2\in (\mathfrak {g}_2\otimes \Omega _2)^0\)  such that \((a_0,a_1,a_2)\in \operatorname{Tot}^0_{TW}(\mathfrak {g}^{\Delta _{[0,2]}})\) . To see this, write \(a_1(t,dt)= a_1^0(t)+ a_1^{-1}(t)dt\) ; then a possible choice for \(a_2\)  is
\(a_2(s_0,s_1,ds_0,ds_1)= a^0_2(s_0,s_1)+ a^{-1}_{2,0}(s_0,s_1)ds_0 +a^{-1}_{2,1}(s_0,s_1)ds_1+ a^{-2}_{2}(s_0,s_1)ds_0ds_1,\) 
with
\(a^0_2(s_0,s_1)&=\partial _{1,2}a^0_1(s_0) + \partial _{0,2}a^0_1(s_1) -\partial _{1,2}a^0_1(0) \\&\qquad \qquad +s_1 \frac{\partial _{2,2}a^0_{1}(s_0)-\partial _{1,2}a^0_{1}(s_0)-\partial _{0,2}a^0_{1}(1-s_0)+\partial _{0,2}a^0_{1}(0)}{1-s_0};\\ &\\a^{-1}_{2,0}(s_0,s_1)&=\partial _{1,2}a_1^{-1}(s_0) +\frac{s_1}{1-s_0}\biggl (\partial _{2,2}a_1^{-1}(s_0) -\partial _{1,2}a_1^{-1}(s_0) \\&\qquad \qquad \qquad \qquad \qquad \qquad +\partial _{0,2}a_1^{-1}(1-s_0) - s_0 \partial _{0,2}a_1^{-1}(0)\biggr );\\ & \\a^{-1}_{2,1}(s_0,s_1)&= \partial _{0,2}a_1^{-1}(s_1) ds_1 -s_0\partial _{0,2}a_1^{-1}(0); \\ &\\a^{-2}_{2}(s_0,s_1)&=0.\) 

It is an easy computation to verify that the element \((a_0,a_1,a_2)\)  actually satisfies the face conditions.

Proposition 7.5 
Let \(\mathfrak {g}^{\Delta }\)  be a semicosimplicial DGLA, such that
\(H^{-1}(\mathfrak {g}_2)=0\) . The map \(\Phi :\operatorname{Def}_{{\operatorname{Tot}}_{TW}(\mathfrak {g}^{\Delta _{[0,2]}})}(A) \rightarrow H^1_{\rm sc}(\exp \mathfrak {g}^\Delta )(A)\)  is injective.

Consider the commutative diagram
\({\operatorname{Def}_{\operatorname{Tot}_{TW}(\mathfrak {g}^{\Delta _{[0,2]}})}(A)[r]^-{\operatorname{Id}_{[0,1]}} [d]^-{\Phi }&\operatorname{Def}_{\operatorname{Tot}_{TW}(\mathfrak {g}^{\Delta _{[0,1]}})}(A)[d]_{\cong }^-{\Phi _{[0,1]}} \\ H^1_{\rm sc}(\exp \mathfrak {g}^\Delta ) (A)[r]^-{i} &H^1_{\rm sc}(\exp \mathfrak {g}^{\Delta _{[0,1]}})(A); }\) 
since the map \(\Phi _{[0,1]}\)  is an isomorphism by Proposition
REF , it is sufficient to prove that \(\operatorname{Id}_{[0,1]}\)  is injective.
Let \((x_0,x_1,x_2)\)  and \((x^{\prime }_0, x^{\prime }_1, x^{\prime }_2)\)  be two Maurer-Cartan elements for \(\operatorname{Tot}_{TW}(\mathfrak {g}^{\Delta _{[0,2]}})\) , such that \((x_0,x_1)\) 
and \((x^{\prime }_0, x^{\prime }_1)\)  are gauge equivalent elements in
\(\operatorname{MC}_{\operatorname{Tot}_{TW}(\mathfrak {g}^{\Delta _{[0,1]}})}(A)\) . Let \((a_0,a_1) \in \operatorname{Tot}^0_{TW}(\mathfrak {g}^{\Delta _{[0,1]}})\otimes \mathfrak {m}_A\)  be an element realizing the gauge equivalence between \((x^{\prime }_0,x^{\prime }_1)\) 
and \((x_0, x_1)\) , and let \((a_0,a_1,a_2)\)  be a lift of \((a_0,a_1)\)  in
\(\operatorname{Tot}^0_{TW}(\mathfrak {g}^{\Delta _{[0,2]}})\otimes \mathfrak {m}_A\)  (see Remark REF ). Then \((x^{\prime }_0,x^{\prime }_1,x^{\prime }_2)\)  is gauge equivalent via \((a_0,a_1,a_2)\)  to the Maurer-Cartan element \((x_0,x_1,e^{a_2}*x^{\prime }_2)\)  and we are left to prove
that \((x_0,x_1,e^{a_2}*x^{\prime }_2)\)  is gauge equivalent to \((x_0,x_1,x_2)\) .
To see this, consider the DGLA \(\operatorname{Tot}_{TW}(\mathfrak {g}^{\Delta _{[0,2]}})\otimes \mathfrak {m}_A\)  and modify its
differential with the Maurer-Cartan element \((x_0,x_1,x_2)\) , as in Remark REF . Translation by \((x_0,x_1,x_2)\)  gives an isomorphism
\(\operatorname{Def}_{\operatorname{Tot}_{TW}(\mathfrak {g}^{\Delta _{[0,2]}})}(A)\cong \operatorname{Def}^{(x_0,x_1,x_2)}_{\operatorname{Tot}_{TW}(\mathfrak {g}^{\Delta _{[0,2]}})}(A);\) 
hence \((x_0,x_1,x_2)\)  and \((x_0,x_1,e^{a_2}*x^{\prime }_2)\)  will be gauge equivalent in \(\operatorname{MC}_{\operatorname{Tot}_{TW}(\mathfrak {g}^{\Delta _{[0,2]}})}(A)\)  if and only if \((0,0,0)\)  and \((0,0,e^{a_2}*x^{\prime }_2-x_2)\)  are gauge-equivalent in
\(\operatorname{MC}^{(x_0,x_1,x_2)}_{\operatorname{Tot}_{TW}(\mathfrak {g}^{\Delta _{[0,2]}})}(A)\) .
Next, observe that the sub-DGLA
\(\operatorname{Tot}_{TW}(\mathfrak {g}^{\Delta _{[2,2]}})\otimes \mathfrak {m}_A\)  of \(\operatorname{Tot}_{TW}(\mathfrak {g}^{\Delta _{[0,2]}})\otimes \mathfrak {m}_A\)  is closed under the
modified differential \(d_{(x_0,x_1,x_2)}\) , so we can consider the deformation functor \(\operatorname{Def}^{(x_0,x_1,x_2)}_{\operatorname{Tot}_{TW}(\mathfrak {g}^{\Delta _{[2,2]}})}(A)\) . Since \(H^{1}(\operatorname{Tot}_{TW}(\mathfrak {g}^{\Delta _{[2,2]}}),d_{TW})=H^1(\operatorname{Tot}(\mathfrak {g}^{\Delta _{[2,2]}}),d_{\rm Tot})=H^{-1}(\mathfrak {g}_2)=0\) , this deformation functor is trivial (see Remark REF ). Therefore \((0,0,e^{a_2}*x^{\prime }_2-x_2)\)  is gauge equivalent to \((0,0,0)\)  as an element of \(\operatorname{MC}^{(x_0,x_1,x_2)}_{\operatorname{Tot}_{TW}(\mathfrak {g}^{\Delta _{[2,2]}})}(A)\) , and so, a fortiori, as an element of \(\operatorname{MC}^{(x_0,x_1,x_2)}_{\operatorname{Tot}_{TW}(\mathfrak {g}^{\Delta _{[0,2]}})}(A)\) .
Summing up, and recalling Proposition REF , we have proved:
Theorem 7.6 
Let \(\mathfrak {g}^{\Delta }\)  be a semicosimplicial DGLA, and let
\(\operatorname{Tot}_{TW}(\mathfrak {g}^{\Delta })\)  and \(\operatorname{Tot}_{TW}(\mathfrak {g}^{\Delta _{[0,2]}})\)  be
the Thom-Whitney DGLAs associated with \(\mathfrak {g}^{\Delta }\)  and
\(\mathfrak {g}^{\Delta _{[0,2]}}\) , respectively. Assume that \(H^{-1}(\mathfrak {g}_2)=0\) ;
then, there is a natural isomorphism of funtors
\(\operatorname{Def}_{\operatorname{Tot}_{TW}(\mathfrak {g}^{\Delta _{[0,2]}})} \cong H^1_{\rm sc}(\exp \mathfrak {g}^\Delta )\) . If moreover \(H^{j}(\mathfrak {g}_i)=0\)  for all \(i\ge 0\)  and
\(j<0\) , then there is a natural isomorphism of funtors
\(\operatorname{Def}_{\operatorname{Tot}_{TW}(\mathfrak {g}^{\Delta })} \cong H^1_{\rm sc}(\exp \mathfrak {g}^\Delta )\) . In particular, in this case, the tangent space to
\(H^1_{\rm sc}(\exp \mathfrak {g}^\Delta )\)  is \(H^1({\rm Tot}(\mathfrak {g}^\Delta ))\)  and
obstructions are contained in \(H^2({\rm Tot}(\mathfrak {g}^\Delta ))\) .

Theorem 7.7 Let \(X\)  be a paracompact Hausdorff topological space, and let
\(\mathcal {L}\)  be a sheaf of differential graded Lie algebras on \(X\) , such that the DGLAs \(\mathcal {L}(U_{i_0 \ldots i_k})\)  has no negative cohomology. Then, every
refinement \({\mathcal {V}}\ge {\mathcal {U}}\)  of open covers of
\(X\)  induces a natural morphism of deformation functors
\(\operatorname{Def}_{\operatorname{Tot}_{TW}({\mathcal {L}}({\mathcal {U}}))}\rightarrow \operatorname{Def}_{\operatorname{Tot}_{TW}({\mathcal {L}}({\mathcal {V}}))}\) . In
particular, the direct limit
\(\operatorname{Def}_{[{\mathcal {L}}]} = \lim _{\stackrel{\longrightarrow }{\mathcal {U}}}\operatorname{Def}_{\operatorname{Tot}_{TW}({\mathcal {L}}({\mathcal {U}}))}\) 
is well defined and there is
natural isomorphism of functors of Artin rings
\(H_{\rm Ho}^1(X;\exp {\mathcal {L}})\cong \operatorname{Def}_{[{\mathcal {L}}]}.\) 
Moreover, if acyclic open covers for \(\mathcal {L}\)  are cofinal
in the directed family of all open covers of \(X\) , then
\( H_{\rm Ho}^1(X; \exp \mathcal {L}) \cong H_{\rm sc}^1(\exp \mathcal {L}(\mathcal {U}))\qquad \text{and}\qquad \operatorname{Def}_{[{\mathcal {L}}]}\cong \operatorname{Def}_{\operatorname{Tot}_{TW}({\mathcal {L}}({\mathcal {U}}))},\) 
for every \({\mathcal {L}}\) -acyclic open cover \({\mathcal {U}}\)  of \(X\) .

Let \({\mathcal {V}}\ge {\mathcal {U}}\)  be a refinement of open
covers of \(X\) , and let \(\tau \)  be a refinement function, it induces a natural morphism of semicosimplicial Lie algebras
\({\mathcal {L}}({\mathcal {U}})\rightarrow {\mathcal {L}}({\mathcal {V}})\)  and so a commutative diagram of
natural transformations
\({\operatorname{Def}_{\operatorname{Tot}_{TW}({\mathcal {L}}({\mathcal {U}}))}[d][r]^-{\sim }& H^1_{\rm sc}(\exp {\mathcal {L}}({\mathcal {U}})) [d]\\\operatorname{Def}_{\operatorname{Tot}_{TW}({\mathcal {L}}({\mathcal {V}}))}[r]^-{\sim }&H^1_{\rm sc}(\exp {\mathcal {L}}({\mathcal {V}})).}\) 
Horizontal arrows are isomorphisms by Theorem REF , and the right vertical
arrow is independent of the refinement function \(\tau \) , as
observed in Lemma REF . Hence, also the
left morphism is independent of \(\tau \) , then the direct limit
\(\operatorname{Def}_{[{\mathcal {L}}]} = \lim _{\stackrel{\longrightarrow }{\mathcal {U}}} \operatorname{Def}_{\operatorname{Tot}_{TW}({\mathcal {L}}({\mathcal {U}}))}\) 
is well defined and we have a natural isomorphism
\(\operatorname{Def}_{[{\mathcal {L}}]}\cong H_{\rm Ho}^1(X;\exp {\mathcal {L}})\) .
Assume now that acyclic open covers for \({\mathcal {L}}\)  are
cofinal in the family of all open covers of \(X\) . Then, for any
refinement \({\mathcal {V}}\ge {\mathcal {U}}\)  of acyclic open
covers, the DGLAs-morphism \(\operatorname{Tot}_{TW}({\mathcal {L}}({\mathcal {U}}))\rightarrow \operatorname{Tot}_{TW}({\mathcal {L}}({\mathcal {V}}))\)  is a
quasi-isomorphism by Leray's theorem. Therefore, we have a
commutative diagram of natural transformations
\({\operatorname{Def}_{\operatorname{Tot}_{TW}({\mathcal {L}}({\mathcal {U}}))}[d]^\wr [r]^-{\sim }&H^1_{\rm sc}(\exp {\mathcal {L}}({\mathcal {U}}))[d]\\\operatorname{Def}_{\operatorname{Tot}_{TW}({\mathcal {L}}({\mathcal {V}}))}[r]^-{\sim }&H^1_{\rm sc}(\exp {\mathcal {L}}({\mathcal {V}})),\\ }\) 
where also the right vertical arrow is forced to be an
isomorphism. Taking the direct limit over \({\mathcal {L}}\) -acyclic
covers, we obtain that, if \({\mathcal {U}}\)  is an \({\mathcal {L}}\) -acyclic open cover of \(X\) , then \( H_{\rm Ho}^1(X; \exp \mathcal {L}) \cong H^1_{\rm sc}(\exp \mathcal {L}(\mathcal {U}))\)  and \(\operatorname{Def}_{[{\mathcal {L}}]}\cong \operatorname{Def}_{\operatorname{Tot}_{TW}({\mathcal {L}}({\mathcal {U}}))}\) .

Conclusions and further developements
We can now sum up our results to obtain a DGLA description of
infinitesimal deformations of a coherent sheaf. In Section
, we analised infinitesimal deformations of a
coherent sheaf \(\mathcal {F}\)  of \(\mathcal {O}_X\) -modules on a ringed space
\((X,\mathcal {O}_X)\) . If \(\mathcal {E}^{\cdot } \rightarrow \mathcal {F}\rightarrow 0\)  is
a locally free resolution of \(\mathcal {F}\)  on \(X\) , we showed how
infinitesimal deformations of \({\mathcal {F}}\)  can be expressed in terms
of the sheaf of DGLAs \(\mathcal {E}nd^* (\mathcal {E}^{\cdot })\) . More precisely, in
Section , we showed that the functor
of infinitesimal deformations of \({\mathcal {F}}\)  is isomorphic to
\(H^1_{\rm Ho}(X; \exp {\mathcal {E}nd}^*(\mathcal {E}^{\cdot }))\) .
Since negative Ext-groups between coherent sheaves are always trivial,
all terms in the semicosimplicial DGLA \(\mathcal {E}nd^*(\mathcal {E}^{\cdot })(\mathcal {U})\) 
have zero negative cohomology. Therefore,
Theorem REF  applies and we obtain that the functor of
infinitesimal deformations of \({\mathcal {F}}\)  is isomorphic to
\(\operatorname{Def}_{[\mathcal {E}nd^*(\mathcal {E}^{\cdot })]}\) ; in particular, we recover the well
known
fact that the tangent space to \(\operatorname{Def}_{\mathcal {F}}\)  is
\(\operatorname{Ext}^1({\mathcal {F}},{\mathcal {F}})\)  and that its obstructions are
contained in \(\operatorname{Ext}^2({\mathcal {F}},{\mathcal {F}})\) .
Moreover, if \(X\)  is a smooth complex variety, then the DGLA controlling
infinitesimal deformations of \(\mathcal {F}\)  turns out to be not at all
mysterious. Indeed, let \( {\mathcal {E}nd}^*(\mathcal {E}^{\cdot }) \rightarrow \mathcal {A}^{0,*}_X(\mathcal {E}nd^*(\mathcal {E}^{\cdot })) \)  be the Dolbeault
resolution of \({\mathcal {E}nd}^*(\mathcal {E}^{\cdot })\) . Since this resolution is
fine, by Theorem REF  the functor of
infinitesimal deformations of \({\mathcal {F}}\)  is isomorphic to the
deformation functor associated with the DGLA \(A^{0,*}_X(\mathcal {E}nd^*(\mathcal {E}^{\cdot }))\)  of global sections of \(\mathcal {A}^{0,*}_X(\mathcal {E}nd^*(\mathcal {E}^{\cdot }))\) . We can also give an explicit description of this
isomorphism of deformation functors. Indeed, a natural isomorphism
\( \operatorname{Def}_{A^{0,*}_X(\mathcal {E}nd^*(\mathcal {E}^{\cdot }))}(B) \rightarrow \operatorname{Def}_{\mathcal {F}}(B), \qquad \mbox{for}\ B \in \bf {Art}_{\mathbb {K}} \) 
is defined by associating with every Maurer-Cartan element \(\xi \)  of
the DGLA \(A^{0,*}_X(\mathcal {E}nd^*(\mathcal {E}^{\cdot }))\)  the cohomology
sheaf of \(({\mathcal {A}}_X^{0,*}(\mathcal {E}^{\cdot }) \otimes B, \overline{\partial }+d_{\mathcal {E}^{\cdot }}+ \xi )\) . Note that, by semicontinuity, this cohomology sheaf is concentrated in degree zero.
The techniques developed in this paper apply to a wide range of other
geometric examples. More explicitly, we can use them in all cases
when local deformations admit
a simple DGLA description in terms of a resolution of the object to be
deformed, for instance, in the case of infinitesimal deformations of a
singular variety. Namely, let \(X\)  be a singular variety, \({\mathcal {O}}_X\)  the sheaf of regular function of \(X\)  and \({\mathcal {R}}^\cdot \rightarrow {\mathcal {O}}_X\)  its standard free resolution [4]}. Then, the deformation functor of infinitesimal
deformations of \(X\)  is isomorphic to \( H_{\rm Ho}^1(X; \exp {{\mathcal {D}}er}^*({\mathcal {R}}^\cdot ) )\) ; see [21]} for details. From
this, we also recover the classical result that the tangent space to
deformations of \(X\)  is \(\operatorname{Ext}^1({\mathbb {L}}_X,{\mathcal {O}}_X)\) , and
that obstructions are contained in \(\operatorname{Ext}^2({\mathbb {L}}_X,{\mathcal {O}}_X)\) , where \({\mathbb {L}}_X\)  is the cotangent complex of \(X\) .
